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Consider an ideal gas whose entropy is given by S=(n)/(2) bar(r) 5R l ln(U)/(n) 2R len(V)/(n) where n= number of moles R= universal gas constant U= internal energy V= volume and sigma= constant. The specific heat at constant volume is given by (a) (5)/(2)nR (b) (1)/(2)nR (c) (3)/(2)nR (d) (9)/(2)nR H.C.U.-2010?
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Consider an ideal gas whose entropy is given by S=(n)/(2) bar(r) 5R l ...
Solution:
Given,
Entropy (S) = (n/2)σ5R ln(U/n2R) + σnR ln(V/n)
where n = number of moles, R = universal gas constant, U = internal energy, V = volume, and σ = constant.
To find: Specific heat at constant volume (Cv)
We know that,
Cv = (∂U/∂T)v
where U = internal energy, T = temperature, and v = constant volume.
We need to express entropy in terms of temperature and volume to find the specific heat.
Step 1: Express entropy in terms of temperature and volume
Using the relation,
dS = (∂S/∂T)v dT + (∂S/∂V)T dV
We get,
(∂S/∂T)v = (n/2)σ5R (1/U)(∂U/∂T)v
(∂S/∂V)T = σnR (1/V)
Substituting the given values, we get,
(∂S/∂T)v = (n/2)σ5R (1/U)Cv
(∂S/∂V)T = σnR (1/V)
Integrating both sides, we get,
S = (n/2)σ5R ln(U/Cv^(5/2)) + σnR ln(V)
Step 2: Find the internal energy
Differentiating the entropy equation with respect to temperature, we get,
(∂S/∂T)v = (n/2)σ5R (1/U)(∂U/∂T)v
Substituting the given entropy equation, we get,
(∂S/∂T)v = (n/2)σ5R (1/U)Cv
Therefore,
Cv = (∂U/∂T)v = (∂T/∂U)v^(-1)
Using the given entropy equation,
S = (n/2)σ5R ln(U/Cv^(5/2)) + σnR ln(V)
Differentiating with respect to U, we get,
(∂S/∂U)v = (n/2)σ5R (Cv^(5/2)/U)
Substituting the above values, we get,
Cv = (5/2)nR
Therefore, the specific heat at constant volume is (Cv) = (5/2)nR.
Answer: Option (a) (5/2)nR
Given,
Entropy (S) = (n/2)σ5R ln(U/n2R) + σnR ln(V/n)
where n = number of moles, R = universal gas constant, U = internal energy, V = volume, and σ = constant.
To find: Specific heat at constant volume (Cv)
We know that,
Cv = (∂U/∂T)v
where U = internal energy, T = temperature, and v = constant volume.
We need to express entropy in terms of temperature and volume to find the specific heat.
Step 1: Express entropy in terms of temperature and volume
Using the relation,
dS = (∂S/∂T)v dT + (∂S/∂V)T dV
We get,
(∂S/∂T)v = (n/2)σ5R (1/U)(∂U/∂T)v
(∂S/∂V)T = σnR (1/V)
Substituting the given values, we get,
(∂S/∂T)v = (n/2)σ5R (1/U)Cv
(∂S/∂V)T = σnR (1/V)
Integrating both sides, we get,
S = (n/2)σ5R ln(U/Cv^(5/2)) + σnR ln(V)
Step 2: Find the internal energy
Differentiating the entropy equation with respect to temperature, we get,
(∂S/∂T)v = (n/2)σ5R (1/U)(∂U/∂T)v
Substituting the given entropy equation, we get,
(∂S/∂T)v = (n/2)σ5R (1/U)Cv
Therefore,
Cv = (∂U/∂T)v = (∂T/∂U)v^(-1)
Using the given entropy equation,
S = (n/2)σ5R ln(U/Cv^(5/2)) + σnR ln(V)
Differentiating with respect to U, we get,
(∂S/∂U)v = (n/2)σ5R (Cv^(5/2)/U)
Substituting the above values, we get,
Cv = (5/2)nR
Therefore, the specific heat at constant volume is (Cv) = (5/2)nR.
Answer: Option (a) (5/2)nR
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Consider an ideal gas whose entropy is given by S=(n)/(2) bar(r) 5R l ln(U)/(n) 2R len(V)/(n) where n= number of moles R= universal gas constant U= internal energy V= volume and sigma= constant. The specific heat at constant volume is given by (a) (5)/(2)nR (b) (1)/(2)nR (c) (3)/(2)nR (d) (9)/(2)nR H.C.U.-2010?
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Consider an ideal gas whose entropy is given by S=(n)/(2) bar(r) 5R l ln(U)/(n) 2R len(V)/(n) where n= number of moles R= universal gas constant U= internal energy V= volume and sigma= constant. The specific heat at constant volume is given by (a) (5)/(2)nR (b) (1)/(2)nR (c) (3)/(2)nR (d) (9)/(2)nR H.C.U.-2010? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider an ideal gas whose entropy is given by S=(n)/(2) bar(r) 5R l ln(U)/(n) 2R len(V)/(n) where n= number of moles R= universal gas constant U= internal energy V= volume and sigma= constant. The specific heat at constant volume is given by (a) (5)/(2)nR (b) (1)/(2)nR (c) (3)/(2)nR (d) (9)/(2)nR H.C.U.-2010? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider an ideal gas whose entropy is given by S=(n)/(2) bar(r) 5R l ln(U)/(n) 2R len(V)/(n) where n= number of moles R= universal gas constant U= internal energy V= volume and sigma= constant. The specific heat at constant volume is given by (a) (5)/(2)nR (b) (1)/(2)nR (c) (3)/(2)nR (d) (9)/(2)nR H.C.U.-2010?.
Consider an ideal gas whose entropy is given by S=(n)/(2) bar(r) 5R l ln(U)/(n) 2R len(V)/(n) where n= number of moles R= universal gas constant U= internal energy V= volume and sigma= constant. The specific heat at constant volume is given by (a) (5)/(2)nR (b) (1)/(2)nR (c) (3)/(2)nR (d) (9)/(2)nR H.C.U.-2010? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider an ideal gas whose entropy is given by S=(n)/(2) bar(r) 5R l ln(U)/(n) 2R len(V)/(n) where n= number of moles R= universal gas constant U= internal energy V= volume and sigma= constant. The specific heat at constant volume is given by (a) (5)/(2)nR (b) (1)/(2)nR (c) (3)/(2)nR (d) (9)/(2)nR H.C.U.-2010? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider an ideal gas whose entropy is given by S=(n)/(2) bar(r) 5R l ln(U)/(n) 2R len(V)/(n) where n= number of moles R= universal gas constant U= internal energy V= volume and sigma= constant. The specific heat at constant volume is given by (a) (5)/(2)nR (b) (1)/(2)nR (c) (3)/(2)nR (d) (9)/(2)nR H.C.U.-2010?.
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