Podcast
Questions and Answers
What is the equation for the isothermal compressibility coefficient?
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.
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)?
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?
In the equation dH = CP dT, what does CP represent?
Signup and view all the answers
For an ideal gas, the change in enthalpy dH is affected by changes in pressure.
For an ideal gas, the change in enthalpy dH is affected by changes in pressure.
Signup and view all the answers
What is the formula for the change in enthalpy (dH) for an ideal gas?
What is the formula for the change in enthalpy (dH) for an ideal gas?
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?
For an ideal gas in a process where pressure is constant, what is the relationship between CP, CV, and 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)?
What is the formula for the change in internal energy (DU) in terms of specific heat at constant volume (CV) and temperature (DT)?
Signup and view all the answers
What is the total differential of z for a function z=f(x,y)?
What is the total differential of z for a function z=f(x,y)?
Signup and view all the answers
What does the partial derivative (∂z/∂x) show?
What does the partial derivative (∂z/∂x) show?
Signup and view all the answers
What is the expression for the change in height dz?
What is the expression for the change in height dz?
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?
What can be set up to measure the change in pressure (DP) when either temperature (T) or volume (V) are held constant?
Signup and view all the answers
What property indicates an exact differential in the statement (dz)?
What property indicates an exact differential in the statement (dz)?
Signup and view all the answers
What does the cyclic rule for inversion state?
What does the cyclic rule for inversion state?
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
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Related Documents
Description
Learn about partial derivatives and total differentials in calculus, including the concept of total differential and its application in multivariable functions.
