Question

7. Suppose you could suddenly increase the speed of every molecule in a gas by a factor of 2.

a. Would the temperature of the gas increase by a factor of $2^{1/2},2$or $2^2?$Explain.

b. Would the molar specific heat at constant volume change? If so, by what factor? If not, why not?

Short answer

a) If the temperature of the gas increase by a factor of $2^{1/2},2$or $2^2$then the temperature will increase by factor$2^2.$

b) A constant volume will result in the same molar-specific heat.

Step-by-step solution

Concept Introduction (Part a)

An observed gas's temperature is a function of the molecules' average translational kinetic energy.

Explanation (Part a)

Here $v_{\mathrm{rms}}\propto \sqrt{T}$, then $T\propto v_{\mathrm{rms}}^2$so doubling the speed of each molecule (which would double $v_{rms}$) quadruples the temperature, which implies that the temperature increases by the factor of $2^2.$

Final Answer (Part a)

Therefore, the temperature increases by the factor of$2^2.$

Concept Introduction (Part b) 

Molar specific heat at constant volume is the amount of heat needed to raise the temperature of $1mole$of a gas through $1K$at constant volume, $CV.$

Explanation (Part b) 

Temperature increases when molecules double in speed.

It is not dependent on the speed of individual molecules but rather the characteristics of those molecules, e.g. whether they are monoatomic, diatomic etc.

The molar specific heat at constant volume is independent of all of this.

Final Answer (Part b) 

When the volume remains the same, the molar specific heat will remain the same.