Question 78: Properties of Matter

Question

Under isothermal condition, the pressure of a gas is given by $$\mathrm{P}=a \mathrm{~V}^{-3}$$, where $$a$$ is a constant and $$\mathrm{V}$$ is the volume of the gas. The bulk modulus at constant temperature is equal to

Accepted Answer

The bulk modulus ($$B$$) of a substance is defined as the ratio of the infinitesimal pressure increase ($$\Delta P$$) to the relative decrease in volume ($$\frac{-\Delta V}{V}$$) at constant temperature:
$$B = -V \frac{\Delta P}{\Delta V}$$
To find the bulk modulus for the given pressure-volume relationship, we first need to find the differential change in pressure with respect to volume:
$$P = aV^{-3}$$
Differentiate $$P$$ with respect to $$V$$:
$$\frac{dP}{dV} = -3aV^{-4}$$
Now we can use the definition of the bulk modulus:
$$B = -V \frac{\Delta P}{\Delta V} = -V \frac{dP}{dV}$$
Plug in the value for $$\frac{dP}{dV}$$:
$$B = -V(-3aV^{-4})$$
Simplify the expression:
$$B = 3aV^{-3}$$
Notice that $$3aV^{-3}$$ is equal to $$3P$$, since $$P = aV^{-3}$$:
$$B = 3P$$
Therefore, the bulk modulus at constant temperature is equal to 3P.

Answer

$$\frac{P}{2}$$

Answer

2 P

Answer

3 P

Answer

P

Properties of Matter

Under isothermal condition, the pressure of a gas is given by $\mathrm{P}=a \mathrm{~V}^{-3}, where a is a constant and \mathrm{V}$ is the volume of the gas. The bulk modulus at constant temperature is equal to

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