Calculus Chapter 6: Partial Derivatives & Total Differentials

Questions and Answers

What is the equation for the isothermal compressibility coefficient?

k = - (V / P) * (∂P/∂V)T

The ___ as a Function of T & V for an ideal gas is DU = CV DT.

change in internal energy

What is the equation for specific enthalpy (H) in terms of internal energy (U), pressure (P), and volume (V)?

H = U + PV

In the equation dH = CP dT, what does CP represent?

Specific heat at constant pressure Signup and view all the answers

For an ideal gas, the change in enthalpy dH is affected by changes in pressure.

False Signup and view all the answers

What is the formula for the change in enthalpy (dH) for an ideal gas?

dH = CPdT Signup and view all the answers

For an ideal gas in a process where pressure is constant, what is the relationship between CP, CV, and R?

CP = CV + R Signup and view all the answers

What is the formula for the change in internal energy (DU) in terms of specific heat at constant volume (CV) and temperature (DT)?

DU = CVDT Signup and view all the answers

What is the total differential of z for a function z=f(x,y)?

dz = (∂z/∂x)dx + (∂z/∂y)dy Signup and view all the answers

What does the partial derivative (∂z/∂x) show?

How z is modified if x is changed while y is kept constant. Signup and view all the answers

What is the expression for the change in height dz?

dz = (∂z/∂x)dx + (∂z/∂y)dy Signup and view all the answers

What can be set up to measure the change in pressure (DP) when either temperature (T) or volume (V) are held constant?

Both Experiment #1 and Experiment #2 Signup and view all the answers

What property indicates an exact differential in the statement (dz)?

The order of differentiation in the second partial derivatives. Signup and view all the answers

What does the cyclic rule for inversion state?

∂x(∂y/∂z) + ∂y(∂z/∂x) + ∂z(∂x/∂y) = 0 Signup and view all the answers

Study Notes

Partial Derivatives and Total Differentials

For a function z = f(x, y), the total differential of z is:
- dz = (∂z/∂x) dx + (∂z/∂y) dy
Partial derivatives show how z changes if x or y changes, while keeping the other variable constant

State Functions and Their Properties

State functions, such as internal energy (U) and enthalpy (H), have exact differential properties
For an exact differential, the order of differentiation is not important:
- (∂²z/∂x∂y) = (∂²z/∂y∂x)
Cyclic rule: (∂x/∂y) (∂y/∂z) (∂z/∂x) = -1
Inversion: (∂x/∂y) = 1 / (∂y/∂x)

Ideal Gas Equation of State (EOS)

EOS: P = nRT/V
Total differential: dP = (nR/V) dT - (nRT/V²) dV
Partial derivatives: (∂P/∂T) = nR/V, (∂P/∂V) = -nRT/V²

Isobaric Thermal Expansion Coefficient and Isothermal Compressibility

b = (∂V/∂T)P = 1/βV, where β is the isothermal compressibility
k = - (∂V/∂P)T = 1/βV

General EOS

dP = (∂P/∂T) dT + (∂P/∂V) dV
Using inversion: (∂P/∂T) = b, (∂P/∂V) = -kV
EOS: dP = b dT - kV dV

U as a Function of T and V

dU = (∂U/∂T) dT + (∂U/∂V) dV
For an ideal gas: dU = CV dT, U only depends on T
For real gases: U depends on V, but the dependence is weak
For liquids and solids: U depends on V, but dV is very small

H as a Function of T and P

dH = (∂H/∂T) dT + (∂H/∂P) dP
For an ideal gas: dH = CP dT, H only depends on T
From the definition of H: H = U + PV
CP = (∂H/∂T)P = (∂U/∂T)P + P (∂V/∂T)P### Thermodynamic Processes
For a constant temperature process (dT = 0), the equation becomes:
- dH = VdP + PdV

Ideal Gas

For an ideal gas, the internal energy (U) is only a function of temperature (T):
- æç ¶U ö÷ = 0
- V = nRT
The enthalpy (H) is also only a function of temperature (T):
- æç ¶H ö÷ = 0
- H = U + PV
The equation for dH becomes:
- dH = C_P dT

Enthalpy (H) as a Function of T and P

For an ideal gas, H is a function of T only and is not affected by changes in P:
- dH = C_P dT
- DH = ò C_P dT

Liquids and Solids

For liquids and solids, which are not compressible, the equation for dH becomes:
- dH » C_P dT + VdP

CP and CV

The relationship between CP and CV:
- C_P = CV + R
- C_P = CV + nR

Formula Summary

Summary of key formulas:
- dU = CV dT
- dH = C_P dT
- DH = C_P DT
- DH » ò C_P dT + VDP
- C_P = CV + R