Question

The bulk modulus of an ideal gas at constant temperature is:
A. Equal to its pressure
B. Equal to its volume
C. Equal to p/2
D. Cannot be determined

Answer 1

Hint: Recall the concept of the bulk modulus of the ideal gas at the constant temperature. Hence, also recall the concept of isothermal bulk modulus of the ideal gas. So, check the value of the isothermal bulk modulus of the ideal gas whether it is equal to the pressure, volume or half of the pressure on the gas.

Complete answer:
The bulk modulus $$K$$ of the material of the substance is given by
$$K={\displaystyle \frac{\mathrm{\Delta }P}{{\displaystyle \frac{\mathrm{\Delta }V}{V}}}}$$ …… (1)
Here, $$P$$ is the pressure on the material, $$\mathrm{\Delta }V$$ is the change in volume of the substance and $$V$$ is the original volume of the substance.

The ideal gas equation is given by
$$PV=nRT$$
Here, $$P$$ is the pressure, $$V$$ is the volume, $$n$$ is the number of moles of the gas molecules, $$R$$ is the gas constant and $$T$$ is the temperature.
Let us assume that the temperature of the gas is constant.
Differentiate the above equation.
$$P\mathrm{\Delta }V+V\mathrm{\Delta }P=0$$
$$\Rightarrow {\displaystyle \frac{\mathrm{\Delta }V}{V}}=-{\displaystyle \frac{\mathrm{\Delta }P}{P}}$$
Substitute $$-{\displaystyle \frac{\mathrm{\Delta }P}{P}}$$ for $${\displaystyle \frac{\mathrm{\Delta }V}{V}}$$ in equation (1).
$$K={\displaystyle \frac{\mathrm{\Delta }P}{-{\displaystyle \frac{\mathrm{\Delta }P}{P}}}}$$
$$\therefore K=-P$$
The negative sign indicates that the gas is compressed when the pressure is applied. The bulk modulus of an ideal gas at constant temperature is known as the isothermal bulk modulus of the ideal gas. This isothermal bulk modulus of the ideal gas is equal to pressure on the gas.

Hence, the correct option is A.

Note:The students may get confused between the bulk modulus of the ideal gas at constant temperature and the bulk modulus of the material of any substance. These two are different physical quantities and hence do not have the same value. So, the students should perfectly keep in mind these two different concepts.