Question

For hydrogen gas ${{C}_{p}}-{{C}_{v}}=a$ and for oxygen gas ${{C}_{p}}-{{C}_{v}}=b$ then the relation between a and b is(where ${{C}_{p}}$ and ${{C}_{v}}$ are gram specific heats).
A. a=16b
B. b=16b
C. a=b
D. none of these.

Answer 1

The specific heat of the substance is defined as the amount of heat requires raising the temperature with unit mass through one degree centigrade. ${{C}_{p}}$ is the quantity of heat required to raise the temperature of 1g of a gas through 1℃ at constant volume, also ${{C}_{v}}$ is the amount of heat required to raise the temperature of 1 mole of a gas through 1℃ at constant pressure.

Complete step-by-step answer:
The specific heat of a gas at constant pressure, defined as (${{C}_{p}}$)is the quantity of heat required to raise the temperature of 1g of a gas through 1℃ at constant volume.
The molar specific heat of a gas at constant pressure, (${{C}_{v}}$) is the amount of heat required to raise the temperature of 1 mole of a gas through 1℃ at constant pressure.
${{C}_{p}}$is greater than ${{C}_{v}}$, in case of ${{C}_{p}}$the whole amount of heat is used to raise the temperature of a gas through 1℃, while in other case heat is supposed not to do heating only but also use to do work for the expansion of gas.
Now, the relation between specific heats is:
${{C}_{p}}-{{C}_{v}}=r=R/M(m=molecular\ mass)$
Now for hydrogen gas:
$\begin{aligned} & {{C}_{p}}-{{C}_{v}}=a \\\ & also,a=R/M \\\ \end{aligned}$
$\begin{aligned} & a=R/M \\\ & a=R/2(M=2,for\ hydrogen) \\\ \end{aligned}$
Now for oxygen gas:
$\begin{aligned} & {{C}_{p}}-{{C}_{v}}=b \\\ & \Rightarrow b=R/M \\\ & b=R/32(M=32,for\ oxygen) \\\ \end{aligned}$
Now, we will find the relation between a and b:
$\begin{aligned} & (a/b)=(R/2)/(R/32) \\\ & (a/b)=2/32 \\\ & (a/b)=1/16 \\\ \end{aligned}$
Hence, we find that the relation between a and b is, a=16b.

So, the correct answer is “Option A”.

Note: The specific heat of a gas at constant pressure is greater than specific heat at constant volume. The specific heat of a substance is defined as the amount of heat required to raise the temperature of 1g of a gas through 1℃. Also, we use M as the molecular weight of the gas in the difference of two specific heats.