## Saturday, April 28, 2007

### Two Questions on Specific Heat Capacity of a Gas

Questions similar to the following one are often found in Medical and Engineering entrance examination question papers:

One mole of an ideal mono atomic gas is mixed with two moles of an ideal diatomic gas. The ratio of specific heats of the mixture is

(a) 1.5 (b) 1.4 (c) 10/6 (d) 15/11 (e) 19/13

You should remember that the values of molar specific heats at constant volume Cv) for mono atomic and diatomic gases are respectively (3/2)R and (5/2)R where R is universal gas constant. The values of molar specific heat at constant pressure Cp) are therefore (5/2)R and (7/2)R respectively, in accordance with Meyer’s relation [Cp – Cv =R].

Therefore, Cv of the mixture = [1×(3/2)R + 2×(5/2)R]/(1+2) = (13/6)R

Cp of the mixture = Cv + R = (19/6)R.

Ratio of specific heats of the mixture, γ = Cp/Cv = 19/13.

[Generally, if n1 moles of a gas having ratio of specific heats γ1 is mixed with n2 moles of a gas having ratio of specific heats γ2, the ratio of specific heats of the mixture is given by the relation, (n1+ n2)/(γ– 1) = n1/( γ1 –1) + n2/( γ2 –1). You can easily arrive at this result].

If one mole of an ideal mono atomic gas is mixed with one mole of an ideal diatomic gas, the ratio of specific heats of the mixture is 1.5. As an exercise, check this.

Now consider the following MCQ:

The following sets of experimental values of Cv and Cp of a given sample of gas were reported by five groups of students. The unit used is calorie mole1 K1. Which set gives the most reliable values?

(a) Cv = 3, Cp = 4.5 (b) Cv = 2, Cp = 4 (c) Cv = 3, Cp = 4.9 (d) Cv = 2.5, Cp = 4.5 (e) Cv = 3, Cp = 4.2

Since the minimum value of Cv is (3/2)R which is the value for a mono atomic gas, when you express it in calorie mole1 K1, the minimum value is approximately 3. [R = 8.3 J mole1 K1 = 2 calorie mole1 K1, approximately]. Options (b) and (d) are therefore not acceptable. Out of the remaining three options, (c) is the most reliable since Cp – Cv = R, which should be 2 calorie mole1 K1 very nearly.