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Chemistry · Chemical Kinetics

Rate of a Reaction (Average & Instantaneous)

Thermodynamics tells us whether a reaction can happen and equilibrium tells us how far it goes, but neither tells us how fast. The rate of a reaction — the central idea of NCERT Class 12 Chemistry §3.1 — quantifies exactly this: the change in concentration of a reactant or product per unit time. This subtopic builds the vocabulary every later kinetics topic depends on, and NEET examiners reliably test the average-versus-instantaneous distinction, the negative sign, the unique rate, and graphical slopes.

What the Rate of a Reaction Means

The speed of an automobile is expressed as the distance it covers in a certain period of time. By exact analogy, the rate of a reaction is the change in concentration of a reactant or product in unit time. Some reactions are almost instantaneous — the precipitation of $\ce{AgCl}$ when $\ce{AgNO3}$ and $\ce{NaCl}$ solutions are mixed — while others, such as the rusting of iron, crawl over months. Chemical kinetics is the branch that puts a number on this speed.

Because a reactant is consumed while a product accumulates, the rate can be measured in two equivalent ways:

Basis of measurementWhat is observedSign behaviour
Rate of decrease of a reactantConcentration of any one reactant falls with timeChange is negative; multiplied by −1
Rate of increase of a productConcentration of any one product rises with timeChange is positive; no correction needed

Both routes describe the same chemical event, so for a simple reaction they give the same numerical value. The framework below makes this precise.

Rate in Terms of Reactants and Products

Consider the hypothetical reaction $\ce{R -> P}$, assuming the volume of the system stays constant so that concentration and amount track each other. One mole of reactant $\ce{R}$ produces one mole of product $\ce{P}$. Let $[\text{R}]_1$ and $[\text{P}]_1$ be the concentrations at time $t_1$, and $[\text{R}]_2$ and $[\text{P}]_2$ at time $t_2$. Then:

$$\Delta t = t_2 - t_1, \qquad \Delta[\text{R}] = [\text{R}]_2 - [\text{R}]_1, \qquad \Delta[\text{P}] = [\text{P}]_2 - [\text{P}]_1$$

The square brackets denote molar concentration. The rate of disappearance of $\ce{R}$ and the rate of appearance of $\ce{P}$ are written:

$$\text{Rate of disappearance of R} = -\frac{\Delta[\text{R}]}{\Delta t}, \qquad \text{Rate of appearance of P} = +\frac{\Delta[\text{P}]}{\Delta t}$$

NEET Trap

The missing minus sign

Since $\Delta[\text{R}]$ is negative — reactant concentration is falling — it is multiplied by $-1$ so that the reported rate stays a positive quantity. A rate is never negative. Forgetting this sign on the reactant term is the single most common slip when students set up rate expressions.

Rule: put a minus sign on every reactant term, none on product terms; the value you report must be positive.

Average Rate of a Reaction

The two expressions above, taken over a finite interval $\Delta t$, define the average rate $r_{av}$:

$$r_{av} = -\frac{\Delta[\text{R}]}{\Delta t} = +\frac{\Delta[\text{P}]}{\Delta t}$$

Average rate depends on the change in concentration over a chosen stretch of time. It answers "how fast, on average, between this clock reading and that one" — not "how fast right now". A classic NCERT illustration is the hydrolysis of butyl chloride, $\ce{C4H9Cl + H2O -> C4H9OH + HCl}$, whose measured concentrations give a steadily falling average rate.

$[\ce{C4H9Cl}]$ at $t_1$ / mol L⁻¹$[\ce{C4H9Cl}]$ at $t_2$ / mol L⁻¹$r_{av}\times10^{4}$ / mol L⁻¹ s⁻¹
0.1000.09051.90
0.09050.08201.70
0.08200.07411.58
0.06710.05491.22
0.04390.03351.04
0.02100.0170.4

The average rate falls from $1.90\times10^{-4}$ to $0.4\times10^{-4}\ \text{mol L}^{-1}\text{s}^{-1}$ as the reaction proceeds, because the reactant is steadily being used up. This decline is exactly why an average rate cannot describe the rate at any single instant — over the interval it is computed for, it is treated as constant.

Worked Example

For $\ce{R -> P}$, the concentration of the reactant changes from 0.03 M to 0.02 M in 25 minutes. Find the average rate in mol L⁻¹ min⁻¹ and mol L⁻¹ s⁻¹.

$r_{av} = -\dfrac{\Delta[\text{R}]}{\Delta t} = -\dfrac{(0.02 - 0.03)}{25}\ \text{mol L}^{-1}\text{min}^{-1} = \dfrac{0.01}{25} = 4\times10^{-4}\ \text{mol L}^{-1}\text{min}^{-1}.$

Converting to seconds ($1\ \text{min} = 60\ \text{s}$): $r_{av} = \dfrac{4\times10^{-4}}{60} = 6.67\times10^{-6}\ \text{mol L}^{-1}\text{s}^{-1}.$

Instantaneous Rate of a Reaction

To express the rate at a particular moment, we shrink the time interval until it is infinitesimally small — that is, let $\Delta t \to 0$. The result is the instantaneous rate $r_{inst}$, written as a derivative:

$$r_{inst} = -\frac{d[\text{R}]}{dt} = +\frac{d[\text{P}]}{dt}$$

Where average rate uses finite differences $\Delta$, instantaneous rate uses the differential $d$. Physically, $r_{inst}$ is the rate "right now", at one tick of the clock. For the butyl chloride hydrolysis, NCERT reports instantaneous rates that themselves decrease as the reaction advances — for instance $1.22\times10^{-4}$ at $t = 250\ \text{s}$, $1.0\times10^{-4}$ at $t = 350\ \text{s}$, $6.4\times10^{-5}$ at $t = 450\ \text{s}$, and $5.12\times10^{-5}\ \text{mol L}^{-1}\text{s}^{-1}$ at $t = 600\ \text{s}$ — confirming that the reaction is fastest at the start.

Build on this

Once you can measure a rate, the next question is what makes it change. See Factors Influencing the Rate of a Reaction for concentration, temperature and catalyst effects.

Graphical Determination by Tangent Slope

The instantaneous rate is found graphically by plotting concentration against time and drawing a tangent to the curve at the chosen instant; the slope of that tangent is the instantaneous rate. The two figures below contrast a reactant curve (falling) and a product curve (rising), and the role of chords versus tangents.

Figure 1 [R] / mol L⁻¹ time → chord = average rate tangent = instantaneous rate t₁ t₂
Reactant concentration vs time. The dashed purple chord between $t_1$ and $t_2$ gives the average rate ($-\Delta[\text{R}]/\Delta t$); the red tangent at a single point gives the instantaneous rate ($-d[\text{R}]/dt$). The reactant slope is negative, so a minus sign converts it to a positive rate.
Figure 2 [P] / mol L⁻¹ time → slope = +d[P]/dt Δt Δ[P]
Product concentration vs time. The curve rises and flattens as reactant runs out. The tangent at any instant has a positive slope equal to $+d[\text{P}]/dt$, the instantaneous rate of formation; the rise $\Delta[\text{P}]$ over run $\Delta t$ gives its magnitude.

Read the two figures together and the geometry is clear: average rate is the slope of a secant chord joining two points, while instantaneous rate is the slope of the tangent line at a single point. Because the curve bends, no two tangents have the same slope — which is precisely why instantaneous rate is a function of time, not a single fixed number.

Stoichiometric Coefficients & the Unique Rate

For a reaction where every stoichiometric coefficient is one, such as $\ce{Hg(l) + Cl2(g) -> HgCl2(s)}$, the rate of disappearance of each reactant equals the rate of appearance of the product:

$$\text{Rate} = -\frac{\Delta[\ce{Hg}]}{\Delta t} = -\frac{\Delta[\ce{Cl2}]}{\Delta t} = +\frac{\Delta[\ce{HgCl2}]}{\Delta t}$$

But when coefficients differ, the species are consumed and formed at different speeds, so the bare rates no longer agree. Take $\ce{2HI(g) -> H2(g) + I2(g)}$: two moles of $\ce{HI}$ vanish for every one mole of $\ce{H2}$ formed, so $\ce{HI}$ disappears twice as fast as $\ce{H2}$ appears. To make all the expressions give one common value, each rate is divided by its own stoichiometric coefficient. This single agreed value is the unique rate of reaction:

$$\text{Rate} = -\frac{1}{2}\frac{\Delta[\ce{HI}]}{\Delta t} = +\frac{\Delta[\ce{H2}]}{\Delta t} = +\frac{\Delta[\ce{I2}]}{\Delta t}$$

The same recipe extends to any balanced equation. For the bromate–bromide reaction $\ce{5Br^- + BrO3^- + 6H^+ -> 3Br2 + 3H2O}$:

$$\text{Rate} = -\frac{1}{5}\frac{\Delta[\ce{Br^-}]}{\Delta t} = -\frac{\Delta[\ce{BrO3^-}]}{\Delta t} = -\frac{1}{6}\frac{\Delta[\ce{H^+}]}{\Delta t} = +\frac{1}{3}\frac{\Delta[\ce{Br2}]}{\Delta t} = +\frac{1}{3}\frac{\Delta[\ce{H2O}]}{\Delta t}$$

NEET Trap

Coefficient in the numerator vs the denominator

For the general reaction $\ce{aA + bB -> cC + dD}$, the unique rate divides by the coefficient: $-\frac{1}{a}\frac{d[\text{A}]}{dt}$. A frequent error is multiplying instead of dividing. If $\ce{N2}$ disappears at a certain rate in $\ce{N2 + 3H2 -> 2NH3}$, then $\ce{H2}$ disappears three times faster and $\ce{NH3}$ forms twice as fast — read the coefficients off the balanced equation every time.

Rule: faster species have larger coefficients; dividing each rate by its coefficient equalises them into one unique rate.

Worked Example

In the reaction $\ce{2A -> Products}$, the concentration of A decreases from 0.5 mol L⁻¹ to 0.4 mol L⁻¹ in 10 minutes. Calculate the rate of reaction during this interval.

The rate of disappearance of A is $-\dfrac{\Delta[\text{A}]}{\Delta t} = -\dfrac{(0.4 - 0.5)}{10} = 1\times10^{-2}\ \text{mol L}^{-1}\text{min}^{-1}.$

Because the coefficient of A is 2, the unique rate of reaction is $-\dfrac{1}{2}\dfrac{\Delta[\text{A}]}{\Delta t} = \dfrac{1}{2}\times1\times10^{-2} = 5\times10^{-3}\ \text{mol L}^{-1}\text{min}^{-1}.$

Units of Rate of a Reaction

From the defining expressions, rate always carries units of concentration divided by time. The exact form depends on how concentration is measured:

Concentration expressed asTime unitUnit of rateWhere used
Molarity (mol L⁻¹)secondmol L⁻¹ s⁻¹Most solution-phase reactions
Molarity (mol L⁻¹)minutemol L⁻¹ min⁻¹Slower reactions / NCERT problems
Partial pressure (atm)secondatm s⁻¹Gaseous reactions

For a gaseous reaction at constant temperature, concentration is directly proportional to partial pressure, so the rate may be reported as the rate of change of partial pressure of a reactant or product. Note that these units of rate of reaction are independent of the order of the reaction — that is a distinction students often blur with the units of the rate constant.

NEET Trap

Rate units vs rate-constant units

The unit of the rate of a reaction is always concentration time⁻¹ (e.g. mol L⁻¹ s⁻¹), no matter what the order is. The unit of the rate constant $k$, by contrast, changes with the order of the reaction. Examiners exploit this overlap — read the question to see which quantity it asks for.

Rule: rate → mol L⁻¹ s⁻¹ (fixed); $k$ → order-dependent.

Average vs Instantaneous at a Glance

With both quantities defined, the contrast below is the form examiners most often probe. Keep the geometric picture from Figures 1 and 2 attached to each row.

FeatureAverage rate ($r_{av}$)Instantaneous rate ($r_{inst}$)
Time scopeOver a finite interval $\Delta t$At one instant ($\Delta t \to 0$)
Mathematical form$-\dfrac{\Delta[\text{R}]}{\Delta t}$ (finite difference)$-\dfrac{d[\text{R}]}{dt}$ (derivative)
Graphical meaningSlope of the chord between two pointsSlope of the tangent at one point
Value as reaction proceedsConstant for the chosen intervalChanges continuously, generally decreasing
Best used toSummarise speed over a stretch of timeState the rate at a specific moment
Quick Recap

Rate of a Reaction — the essentials

  • Rate of a reaction = change in concentration of a reactant or product per unit time.
  • Reactant terms carry a minus sign so the reported rate stays positive; product terms do not.
  • Average rate $= -\Delta[\text{R}]/\Delta t$ (slope of a chord); instantaneous rate $= -d[\text{R}]/dt$ (slope of a tangent as $\Delta t \to 0$).
  • The instantaneous rate is found by drawing a tangent to the concentration–time curve and taking its slope.
  • Divide each species' rate by its stoichiometric coefficient to get the single unique rate; e.g. $\ce{2HI -> H2 + I2}$ gives $-\tfrac12 d[\ce{HI}]/dt = d[\ce{H2}]/dt = d[\ce{I2}]/dt$.
  • Units: mol L⁻¹ s⁻¹ for solutions, atm s⁻¹ for gases — fixed regardless of order.

NEET PYQ Snapshot — Rate of a Reaction

Direct NEET questions on rate definition and stoichiometric rate are rare; the bank for this subtopic largely tests rate dependence on concentration. Below is the closest genuine PYQ plus a concept card for the core definition.

NEET 2023 · Q.61

For a certain reaction, the rate $= k[\text{A}]^2[\text{B}]$. When the initial concentration of A is tripled keeping concentration of B constant, the initial rate would

  • (1) increase by a factor of three
  • (2) decrease by a factor of nine
  • (3) increase by a factor of six
  • (4) increase by a factor of nine
Answer: (4) increase by a factor of nine

The rate is the instantaneous rate of the reaction. With $r = k[\text{A}]^2[\text{B}]$, replacing $[\text{A}]$ by $[3\text{A}]$ gives $r' = k(3[\text{A}])^2[\text{B}] = 9\,k[\text{A}]^2[\text{B}] = 9r$. The rate of reaction therefore increases nine-fold — a direct application of how concentration controls reaction rate.

Concept

For the reaction $\ce{2HI(g) -> H2(g) + I2(g)}$, which expression correctly gives the unique rate of the reaction?

  • (1) $-2\dfrac{d[\ce{HI}]}{dt}$
  • (2) $-\dfrac{1}{2}\dfrac{d[\ce{HI}]}{dt}$
  • (3) $+\dfrac{1}{2}\dfrac{d[\ce{H2}]}{dt}$
  • (4) $-\dfrac{d[\ce{HI}]}{dt}$
Answer: (2)

$\ce{HI}$ has coefficient 2, so its rate is divided by 2 and carries a minus sign (reactant). Thus the unique rate $= -\tfrac{1}{2}\,d[\ce{HI}]/dt = +d[\ce{H2}]/dt = +d[\ce{I2}]/dt$, making option (2) correct. Option (3) wrongly halves the product term whose coefficient is one.

FAQs — Rate of a Reaction

The conceptual questions NEET aspirants ask most about average and instantaneous rate.

What is the difference between average rate and instantaneous rate of a reaction?
Average rate is the change in concentration of a reactant or product divided by the time interval over which the change occurs, so it represents the rate over a finite stretch of time. Instantaneous rate is the rate at one particular moment, obtained as the time interval shrinks to zero, i.e. the derivative −d[R]/dt or d[P]/dt. Graphically, average rate is the slope of the chord joining two points on the concentration–time curve, while instantaneous rate is the slope of the tangent drawn at a single point.
Why is a negative sign used when expressing rate in terms of a reactant?
The concentration of a reactant decreases with time, so Δ[R] is a negative quantity. Since rate of a reaction must always be a positive number, the change in reactant concentration is multiplied by −1. For products, concentration increases, Δ[P] is positive, and no extra sign is required.
What is the unique rate of a reaction and why divide by stoichiometric coefficients?
When stoichiometric coefficients are not all equal to one, the rate of disappearance of a reactant and the rate of appearance of a product differ numerically. Dividing each rate by its own stoichiometric coefficient makes all the expressions equal, giving a single value called the unique rate of reaction. For 2HI → H2 + I2 the unique rate is −(1/2)d[HI]/dt = d[H2]/dt = d[I2]/dt.
What are the units of rate of a reaction?
Rate has units of concentration per unit time. When concentration is in mol L⁻¹ and time in seconds, the unit is mol L⁻¹ s⁻¹. For gaseous reactions where concentration is expressed through partial pressure, the unit becomes atm s⁻¹. These units are the same for average rate, instantaneous rate and the unique rate of reaction.
How is the instantaneous rate determined graphically?
Plot the concentration of a reactant or product against time. To find the instantaneous rate at a chosen time, draw a tangent that just touches the curve at that point and measure its slope. For a reactant curve the slope is negative, so the instantaneous rate is the negative of that slope; for a product curve the slope is directly the instantaneous rate.
Does the rate of a reaction stay constant during the reaction?
For most reactions the rate is highest at the start, when reactant concentration is greatest, and falls as the reaction proceeds. This is why the concentration–time curve is steep at the beginning and flattens out, and why the instantaneous rate computed from successive tangents keeps decreasing with time.
Related Topics
Chemical Kinetics Back to the full chapter hub and all subtopics. Factors Influencing Rate How concentration, temperature and catalyst change the rate. Rate Law and Order Rate expressions, order, molecularity and rate constant. Integrated Rate Equations Zero- and first-order rate equations and half-life.