Let U=f(P,V,T) be the internal energy of a gas that obeys the ideal gas law PV=nRT (n and r constant). Finda.dUdPv andb.dUdTv.

Answers

Answer 1

The dU/dT at constant P and V is simply nR/P.

According to the ideal gas law, PV = nRT, so we can write P = nRT/V. Using this relationship, we can express the internal energy U as a function of P, V, and T:

U = f(P,V,T) = f(nRT/V, V, T)

To find dU/dP at constant V and T, we can use the chain rule:

dU/dP = (∂U/∂P)V,T + (∂U/∂V)P,T(dP/dP)V,T + (∂U/∂T)P,V(dT/dP)V,T

Since V and T are being held constant, we can simplify the second and third terms to just 0:

dU/dP = (∂U/∂P)V,T

To find (∂U/∂P)V,T, we can differentiate f(nRT/V, V, T) with respect to P, keeping V and T constant:

(∂U/∂P)V,T = (∂f/∂P)nRT/V(-nRT/V²) = -nRT/V²

So, dU/dP at constant V and T is simply -nRT/V².

To find dU/dT at constant P and V, we can again use the chain rule:

dU/dT = (∂U/∂T)P,V + (∂U/∂V)P,T(dV/dT)P,V + (∂U/∂P)V,T(dP/dT)P,V

Since P and V are being held constant, we can simplify the third term to just 0:

dU/dT = (∂U/∂T)P,V + (∂U/∂V)P,T(dV/dT)P,V

To find (∂U/∂T)P,V, we can differentiate f(nRT/V, V, T) with respect to T, keeping P and V constant:

(∂U/∂T)P,V = (∂f/∂T)nRT/V(1) = nR/V

To find (∂U/∂V)P,T, we can differentiate f(nRT/V, V, T) with respect to V, keeping P and T constant:

(∂U/∂V)P,T = (∂f/∂V)nRT/V(-nRT/V²) + (∂f/∂V)V,T = nRT/V² - nRT/V² = 0

Since the ideal gas law shows that PV = nRT, we can write V = nRT/P. Using this relationship, we can simplify the second term of dU/dT to just:

dU/dT = (∂U/∂T)P,V = nR/P

So, dU/dT at constant P and V is simply nR/P.

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Answer 2

a. To find dU/dPv, we need to differentiate U with respect to both P and V while treating T as a constant. Using the chain rule, we have:

dU/dPv = (∂U/∂P)v + (∂U/∂V)p * (dV/dP)v

Since U is a function of P, V, and T, we can express it as U(P,V,T). Using the ideal gas law, we substitute P = nRT/V into U:

U = f(P,V,T) = f(nRT/V, V, T)

Differentiating U with respect to P while treating V and T as constants, we get (∂U/∂P)v = -nRT/V².

Similarly, differentiating U with respect to V while treating P and T as constants, we get (∂U/∂V)p = nRT/V.

Hence, dU/dPv = -nRT/V² + nRT/V * (dV/dP)v.

b. To find dU/dTv, we differentiate U with respect to both T and V while treating P as a constant. Using the chain rule:

dU/dTv = (∂U/∂T)v + (∂U/∂V)t * (dV/dT)v

Differentiating U with respect to T while treating V and P as constants, we get (∂U/∂T)v = (∂f/∂T)v.

Similarly, differentiating U with respect to V while treating T and P as constants, we get (∂U/∂V)t = (∂f/∂V)t.

Hence, dU/dTv = (∂f/∂T)v + (∂f/∂V)t * (dV/dT)v.

Note: The specific form of the function f(P,V,T) is not provided, so we cannot determine the exact values of (∂f/∂T)v, (∂f/∂V)t, and (dV/dT)v without additional information.

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Answer:

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Answer:

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