Kirch off Equation- Introduction, Formula, Practice Problems & FAQs

India's desert region is a unique tourist destination. You will undoubtedly be impressed by your visit and have a once-in-a-lifetime experience.

Days will be exceedingly hot in deserts and nights are much cooler. Do you know, why?

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Sand works well as an absorbent. The nights will be cool there because a good absorber also makes a good emitter. This explains why desert days are hot and desert nights are freezing.

Kirchoff's law illustrates this characteristic of sand, and by understanding this concept, many more aspects of daily life can be simply understood. Therefore, let's conduct a brief examination of this law.

Table of Contents

Introduction to Kirchoff’s Equation
Practice Problems
Frequently Asked Questions - FAQs

Introduction to Kirchoff’s Equation

The enthalpy of a reaction depends on the temperature and this dependence was described using a law known as Kirchhoff's law. The enthalpy of a reaction increases with the increase in the temperature. This can be shown by deriving Kirchoff’s equation.

Kirchhoff’s equation relates the enthalpies at two different temperatures.

Consider a chemical reaction;

$A \to B$

HA & UA are the enthalpy and internal energy of reactant A respectively.

HB & UB are the enthalpy and internal energy of product B respectively.

Thus enthalpy of reaction would be;

$Δ H_{r} = H_{B} - H_{A} . . . . (i)$

Enthalpy is a function of pressure and temperature. At constant pressure, the enthalpy of reaction i.e., equation (i) can be differentiated with respect to temperature (T).

${[\frac{d (Δ H_{r})}{d T}]}_{P} = {[\frac{d H_{B}}{d T}]}_{P} - {[\frac{d H_{A}}{d T}]}_{P} . . . . (i i)$

We know that;

$Δ H_{r} = C_{P} ΔT$

Where CP is heat capacity at constant pressure.

So, it can be written as for the above reaction:

$\frac{d (Δ H_{r})}{d T} = {ΔC}_{P} . . . . (i i i)$

Where, CP is the difference between heat capacities of products and reactants.

$Δ C_{P} = (C_{P})_{B} - (C_{P})_{A} . . . . (i v)$

(CP)B is the heat capacity of a product at constant pressure and (CP)A is the heat capacity of a reactant at constant pressure.

From equation (ii), (iii) & (iv), we get;

$\frac{d (Δ H_{r})}{d T} = (C_{P})_{B} - (C_{P})_{A}$

$d Δ H_{r} = Δ C_{P} d T . . . . (v)$

Equation (v) is Kirchoff's equation.

When CP is constant with respect to temperature and it is raised from T1 to T2.

From the Kirchoff’s equation;

$d Δ H_{r} = Δ C_{P} d T$

Integrating both sides;

$\int_{H_{1}}^{H_{2}} d (Δ H_{r}) = \int_{T_{1}}^{T_{2}} Δ C_{P} d T$

$\Rightarrow Δ H_{2} - Δ H_{1} = Δ C_{P} (T_{2} - T_{1})$

Similarly, we can write;

$\Rightarrow Δ U_{2} - Δ U_{1} = Δ C_{V} (T_{2} - T_{1})$

Practice Problems

Q1. Which of the following Kirchoff’s equations is incorrect?

a. $Δ H_{2} - Δ H_{1} = Δ C_{P} (T_{2} - T_{1})$
b. $Δ U_{2} - Δ U_{1} = Δ C_{V} (T_{2} - T_{1})$
c. $d Δ H_{r} = Δ C_{P} d T$
d. $Δ H_{2} - Δ H_{1} = Δ C_{V} (T_{2} - T_{1})$

Answer: (D)

All options A, B and C are the form of Kirchoff’s equation. Option D can not be correct as instead CV it should be CP. So, H2-H1= CV(T2-T1 ) equation is incorrect.

Q2. For a reaction; A (g)2B (g), if (CP)A=x J/K and (CP)B=y J/K. The value of CP of the reaction is:

a. (2x-y) J/K
b. (2y-x) J/K
c. (x-2y) J/K
d. (y-2x) J/K

Answer: (B)

CP is the difference between heat capacities of products and reactants.

$Δ C_{P} = \sum (C_{P})_{B} - \sum (C_{P})_{A}$

$Δ C_{P} = (2 y - x) J / K$ $(A s (C_{P})_{A} = x J / K a n d (C_{P})_{B} = y J / K)$

Q3. For a reaction; A (g)2B (g), if (CP)A=10 J/K and (CP)B=20 J/K. The temperature of the reaction is raised from 300 K to 600 K. The difference between the change in enthalpies at these two temperatures will be:

a. 9 kJ
b. 9 J
c. 9000 kJ
d. 30 J

Answer: (A)

CP is the difference between heat capacities of products and reactants.

$Δ C_{P} = \sum (C_{P})_{B} - \sum (C_{P})_{A}$

$Δ C_{P} = (2 \times 20 - 10) J / K = 30 J / K (A s (C_{P})_{A} = 10 J / K a n d (C_{P})_{B} = 20 J / K)$

From the Kirchoff’s equation;

$Δ H_{2} - Δ H_{1} = Δ C_{P} (T_{2} - T_{1})$

$\Rightarrow Δ H_{2} - Δ H_{1} = 30 \times (600 - 300) J$

$\Rightarrow Δ H_{2} - Δ H_{1} = 9000 J = 9 k J$

Q4. For a reaction; 2A (g)B (g), if (CP)A=20 J/K and (CP)B=60 J/K. The difference between the change in enthalpies at these two different temperatures is 80 J. The change in the temperature of the reaction would be:

a. 8 K
b. 40 K
c. 400 K
d. 4 K

Answer: (D)

CP is the difference between heat capacities of products and reactants.

$Δ C_{P} = \sum (C_{P})_{B} - \sum (C_{P})_{A}$

$Δ C_{P} = (60 - 2 \times 20) J / K = 20 J / K (A s (C_{P})_{A} = 20 J / K a n d (C_{P})_{B} = 60 J / K)$

From the Kirchoff’s equation;

$Δ H_{2} - Δ H_{1} = Δ C_{P} (T_{2} - T_{1})$

$\Rightarrow (T_{2} - T_{1}) = \frac{Δ H_{2} - Δ H_{1}}{Δ C_{P}}$

$\Rightarrow (T_{2} - T_{1}) = \frac{80 J}{20 J K^{- 1}} = 4 K$

Frequently Asked Questions-FAQs

Q1. Does the enthalpy of a reaction depend on the temperature?
Answer: The enthalpy of the reaction depends on the temperature. This can be proved using Kirchoff’s law and equation. With the increase in the temperature, the enthalpy of the reaction increases considering the heat capacity is constant with the temperature.

Q2. Does change in heat capacity of a reaction at constant pressure varies with temperature?
Answer: The heat capacity of a reaction at constant pressure varies with the temperature. We can get this idea by using Kirchoff’s equation.

$\int d (Δ H_{r}) = \int Δ C_{P} d T$

$Δ H_{r} = \int Δ C_{P} d T + K$ (where, K integration constant)

We cannot directly take the CP term out of the integration as CP varies with temperature. Hence the integration constant appears in the equation.

Q3. Who gave this Kirchoff’s law in Thermodynamics?
Answer: The enthalpy of the reaction depends on the temperature and it is mathematically expressed by G. R. Kirchhoff in 1858 known as Kirchhoff’s equation The equation uses the first law of thermodynamics to derive the equation.