Question

The adiabatic elasticity of a diatomic gas at NTP is:
A. zero
B. $\text{1 }\times \text{ 1}{{\text{0}}^{5}}\text{ N/}{{\text{m}}^{2}}$
C. $\text{1}\text{.4 }\times \text{ 1}{{\text{0}}^{5}}\text{ N/}{{\text{m}}^{2}}$
D. $\text{2}\text{.75 }\times \text{ 1}{{\text{0}}^{5}}\text{ N/}{{\text{m}}^{2}}$

Answer 1

Hint: The adiabatic system is the system in which neither the heat can travel inside the system nor the heat can go outside from the system to the surrounding. The adiabatic elasticity formula is \[{{\text{K}}_{\text{adiabatic}}}\text{ = }\gamma \text{P}\]. Here, gamma is the adiabatic index and P is the pressure.

Complete step by step answer:
- In the question, we have to find the value of adiabatic elasticity for the diatomic gas at NTP.
- Diatomic gas are those species which are made up of two atoms such as hydrogen gas. Whereas monatomic gas is made up of only one atom such as a helium atom.

- Here, NTP means Normal Temperature Pressure in which the value of pressure is \[\text{1}{{\text{0}}^{5}}\]pascal and also we know that the value of gamma for the diatomic gas is 1.4.
- Now, the adiabatic process is the process in which no exchange of the heat between surrounding and system takes place.

- The expression for the adiabatic process is
$\text{P}{{\text{V}}^{\gamma }}\text{ = Constant}$
- By integrating the above equation we will get.
\[{{\text{K}}_{\text{adiabatic}}}\text{ = }\gamma \text{P}\] …..(1)
- So, by putting all the values in equation (1), we will get the value of adiabatic elasticity i.e.
\[{{\text{K}}_{\text{adiabatic}}}\text{ = 1}\text{.4 }\times \text{ 1}{{\text{0}}^{5}}\text{N/}{{\text{m}}^{2}}\]
- So, the value of adiabatic elasticity for the diatomic gas is \[\text{1}\text{.4 }\times \text{ 1}{{\text{0}}^{5}}\text{N/}{{\text{m}}^{2}}\].
So, the correct answer is “Option C”.

Note: The value of gamma for the monatomic gas is 1.67. Gamma or adiabatic index is the ratio of ${{\text{C}}_{\text{P}}}$ to the ${{\text{C}}_{\text{V}}}$. ${{\text{C}}_{\text{V}}}$ is the specific heat at constant volume whereas ${{\text{C}}_{\text{P}}}$ is the specific heat at constant pressure. The value of specific heat at constant pressure is usually greater than the value of specific heat at constant volume that's why the value of gamma is greater than one.