Answer :
Answer:
The coefficient of thermal expansion α is
[tex]\alpha = \frac{1}{T}[/tex]
The coefficient of compressibility
[tex]\beta = \frac{1}{P}[/tex]
Now considering [tex](\frac{ \delta P }{\delta T} )V[/tex]
From equation (1) we have that
[tex]\frac{ \delta P}{\delta T} = \frac{n R }{V}[/tex]
From ideal equation
[tex]nR = \frac{PV}{T}[/tex]
So
[tex]\frac{\delta P}{\delta T} = \frac{PV}{TV}[/tex]
=> [tex]\frac{\delta P}{\delta T} = \frac{P}{T}[/tex]
=> [tex]\frac{\delta P}{\delta T} = \frac{\alpha }{\beta}[/tex]
Explanation:
From the question we are told that
The coefficient of thermal expansion is [tex]\alpha = \frac{1}{V} * (\frac{\delta V}{ \delta P}) P[/tex]
The coefficient of compressibility is [tex]\beta = - (\frac{1}{V} ) * (\frac{\delta V}{ \delta P} ) T[/tex]
Generally the ideal gas is mathematically represented as
[tex]PV = nRT[/tex]
=> [tex]V = \frac{nRT}{P} --- (1)[/tex]
differentiating both side with respect to T at constant P
[tex]\frac{\delta V}{\delta T } = \frac{ n R }{P}[/tex]
substituting the equation above into [tex]\alpha[/tex]
[tex]\alpha = \frac{1}{V} * ( \frac{ n R }{P}) P[/tex]
[tex]\alpha = \frac{nR}{PV}[/tex]
Recall from ideal gas equation [tex]T = \frac{PV}{nR}[/tex]
So
[tex]\alpha = \frac{1}{T}[/tex]
Now differentiate equation (1) above with respect to P at constant T
[tex]\frac{\delta V}{ \delta P} = -\frac{nRT}{P^2}[/tex]
substituting the above equation into equation of [tex]\beta[/tex]
[tex]\beta = - (\frac{1}{V} ) * (-\frac{nRT}{P^2} ) T[/tex]
[tex]\beta =\frac{ (\frac{n RT}{PV} )}{P}[/tex]
Recall from ideal gas equation that
[tex]\frac{PV}{nRT} = 1[/tex]
So
[tex]\beta = \frac{1}{P}[/tex]
Now considering [tex](\frac{ \delta P }{\delta T} )V[/tex]
From equation (1) we have that
[tex]\frac{ \delta P}{\delta T} = \frac{n R }{V}[/tex]
From ideal equation
[tex]nR = \frac{PV}{T}[/tex]
So
[tex]\frac{\delta P}{\delta T} = \frac{PV}{TV}[/tex]
=> [tex]\frac{\delta P}{\delta T} = \frac{P}{T}[/tex]
=> [tex]\frac{\delta P}{\delta T} = \frac{\alpha }{\beta}[/tex]