(4) Partition function of ideal gas comprising N particles of mass m in potential V(r) in equilibrium at constant temperature T (or ẞ = 1/kT) is given by the following:* Z = 1 3N/2 m d³r N! 2лh²ß Let us consider this ideal gas confined in a box 0 ≤ x ≤ L; 0 ≤ y ≤ L; ho ≤ z≤h₁, and in potential V(x, y, z) = mgz (i.e. uniform gravity). (a) Calculate the partition function and Helmholtz free energy F(B, L, ho, h₁). (b) Calculate the pressures at the bottom and the top of the box, po and P₁. (c) Calculate the heat capacity Cy. Your answer should be different from 3/2R of the monoatomic ideal gas. Consider a realistic measurement (such as the size of the box) of the heat capacity using argon gas (molecular weight = 40 [g/mol]), and describe how the difference between your calculated Cy and ³/2R can be reconciled. *h is a number but no knowledge is needed. It's called "Planck's constant" (see lecture notes).
(4) Partition function of ideal gas comprising N particles of mass m in potential V(r) in equilibrium at constant temperature T (or ẞ = 1/kT) is given by the following:* Z = 1 3N/2 m d³r N! 2лh²ß Let us consider this ideal gas confined in a box 0 ≤ x ≤ L; 0 ≤ y ≤ L; ho ≤ z≤h₁, and in potential V(x, y, z) = mgz (i.e. uniform gravity). (a) Calculate the partition function and Helmholtz free energy F(B, L, ho, h₁). (b) Calculate the pressures at the bottom and the top of the box, po and P₁. (c) Calculate the heat capacity Cy. Your answer should be different from 3/2R of the monoatomic ideal gas. Consider a realistic measurement (such as the size of the box) of the heat capacity using argon gas (molecular weight = 40 [g/mol]), and describe how the difference between your calculated Cy and ³/2R can be reconciled. *h is a number but no knowledge is needed. It's called "Planck's constant" (see lecture notes).
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![(4) Partition function of ideal gas comprising N particles of mass m in potential V(r) in equilibrium
at constant temperature T (or ẞ = 1/kT) is given by the following:*
Z =
1
3N/2
m
d³r
N! 2лh²ß
Let us consider this ideal gas confined in a box 0 ≤ x ≤ L; 0 ≤ y ≤ L; ho ≤ z≤h₁, and in
potential V(x, y, z) = mgz (i.e. uniform gravity).
(a) Calculate the partition function and Helmholtz free energy F(B, L, ho, h₁).
(b) Calculate the pressures at the bottom and the top of the box, po and P₁.
(c) Calculate the heat capacity Cy. Your answer should be different from 3/2R of the
monoatomic ideal gas. Consider a realistic measurement (such as the size of the box) of
the heat capacity using argon gas (molecular weight = 40 [g/mol]), and describe how the
difference between your calculated Cy and ³/2R can be reconciled.
*h is a number but no knowledge is needed. It's called "Planck's constant" (see lecture notes).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc158a850-76a9-4504-97b9-8593e0926539%2Fd517dd93-656c-4dc7-8f08-b0a169e51ce1%2Fu22sn7q_processed.png&w=3840&q=75)
Transcribed Image Text:(4) Partition function of ideal gas comprising N particles of mass m in potential V(r) in equilibrium
at constant temperature T (or ẞ = 1/kT) is given by the following:*
Z =
1
3N/2
m
d³r
N! 2лh²ß
Let us consider this ideal gas confined in a box 0 ≤ x ≤ L; 0 ≤ y ≤ L; ho ≤ z≤h₁, and in
potential V(x, y, z) = mgz (i.e. uniform gravity).
(a) Calculate the partition function and Helmholtz free energy F(B, L, ho, h₁).
(b) Calculate the pressures at the bottom and the top of the box, po and P₁.
(c) Calculate the heat capacity Cy. Your answer should be different from 3/2R of the
monoatomic ideal gas. Consider a realistic measurement (such as the size of the box) of
the heat capacity using argon gas (molecular weight = 40 [g/mol]), and describe how the
difference between your calculated Cy and ³/2R can be reconciled.
*h is a number but no knowledge is needed. It's called "Planck's constant" (see lecture notes).
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