Question

An ideal gas is expanding such that PT=constant.The coefficient of volume expansion of the gas is
choices are
a) 1/T
b) 2/T
c) 3/T
d) 4/T

Answer 1

Answer on Question 57145, Physics, Molecular Physics | Thermodynamics

Question:

An ideal gas is expanding such that $PT = const$. The coefficient of volume expansion of the gas is

a) $1 / T$

b) $2 / T$

c) $3 / T$

d) $4 / T$

Solution:

We can find the coefficient of volume expansion of the gas from the formula:

$$\gamma = \frac {1}{V} \frac {d V}{d T}.$$

Let's write the ideal gas law:

$$P V = n R T.$$

From this formula we can find $P$:

$$P = \frac {n R T}{V}.$$

Therefore, substituting $P$ into the formula $PT = const$ we get:

$$\frac {n R T}{V} T = \text{const},$$ $$n R T ^ {2} = \text{const} \cdot V.$$

Let's differentiate the last equation:

$$2 n R T d T = \text{const} \cdot d V,$$ $$\frac {d V}{d T} = \frac {2 n R T}{\text{const}}.$$

Then, we can find the coefficient of volume expansion (from the ideal gas law we obtain $V = \frac{nRT}{P}$ and $PT = const$):

$$\gamma = \frac {1}{V} \frac {d V}{d T} = \frac {2 n R T}{c o n s t \cdot V} = \frac {2 n R T}{p T \cdot \frac {n R T}{P}} = \frac {2}{T}.$$

Answer:

b) $\gamma = 2 / T$.

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