If alpha and beta are the volume and pressure | vedclass

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(A) For an ideal gas, the equation of state is given by $PV = nRT$.
The volume coefficient $\alpha$ is defined as $\alpha = \frac{1}{V} \left( \frac{

Vedclass · Accepted Answer

$\alpha = \beta$

Vedclass · Answer

(A) For an ideal gas, the equation of state is given by $PV = nRT$.
The volume coefficient $\alpha$ is defined as $\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_P$. From $V = \frac{nRT}{P}$, we get $\left( \frac{\partial V}{\partial T} \right)_P = \frac{nR}{P}$. Thus, $\alpha = \frac{1}{V} \left( \frac{nR}{P} \right) = \frac{nR}{PV} = \frac{1}{T}$.
The pressure coefficient $\beta$ is defined as $\beta = \frac{1}{P} \left( \frac{\partial P}{\partial T} \right)_V$. From $P = \frac{nRT}{V}$, we get $\left( \frac{\partial P}{\partial T} \right)_V = \frac{nR}{V}$. Thus, $\beta = \frac{1}{P} \left( \frac{nR}{V} \right) = \frac{nR}{PV} = \frac{1}{T}$.
Comparing the two, we find that $\alpha = \beta = \frac{1}{T}$.

If $\alpha$ and $\beta$ are the volume and pressure coefficients respectively of an ideal gas, then

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