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The entropy sof a system of n particles at a temperature t is given by s=a(nvu)^(1/3) where u and v are internal energy and volume of the system respectively and a is constant.if temperature changes to 4t at constant volume then internal energy of the system becomes (a) four times (b) two times (c) eight times (d) half?
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The entropy sof a system of n particles at a temperature t is given by...
Given:
- Entropy of a system of n particles at temperature t is given by s = a(nvu)^(1/3)
- u and v are the internal energy and volume of the system, respectively
- a is a constant
To find:
- Internal energy of the system when the temperature changes to 4t at constant volume
Solution:
Step 1: Relationship between Entropy and Temperature
The entropy of a system is related to the temperature through the equation:
ΔS = ∫(dQ/T)
Step 2: Entropy Change at Constant Volume
At constant volume, the heat transfer can be written as:
dQ = dU (change in internal energy)
Substituting this in the entropy equation:
ΔS = ∫(dU/T)
Step 3: Calculating the Change in Internal Energy
Using the given expression for entropy:
s = a(nvu)^(1/3)
Taking the derivative of s with respect to u:
ds/du = a(1/3)(nvu)^(-2/3) * nv
Rearranging the equation:
du = (3/(anv)) * (nvu)^(2/3) * ds
Step 4: Calculating the Internal Energy
Substituting the above expression for du in the entropy equation:
ΔS = ∫(dU/T)
ΔS = ∫(3/(anv)) * (nvu)^(2/3) * ds / T
Rearranging the equation:
ΔS = (3/(anvT)) * ∫(nvu)^(2/3) * ds
Integrating both sides:
ΔS = (3/(anvT)) * (3/5) * (nvu)^(5/3) + C
Where C is the constant of integration.
Step 5: Finding the Internal Energy
At constant volume, the change in entropy is given by:
ΔS = ∫(dU/T)
Substituting the expression for entropy:
(3/(anvT)) * (3/5) * (nvu)^(5/3) + C = ∫(dU/T)
Since the volume is constant, dv = 0. Therefore, the integral simplifies to:
(3/(anvT)) * (3/5) * (nvu)^(5/3) + C = ∫(dU/T) = (1/T) * ∫dU
Integrating both sides:
(3/(anvT)) * (3/5) * (nvu)^(5/3) + C = (1/T) * U
Simplifying the equation:
(3/(anvT)) * (3/5) * (nvu)^(5/3) = (1/T) * U - C
Since entropy is given by s = a(nvu)^(1/3), we can substitute this expression:
(3/(anvT)) * (3/5) * s^(5/3) = (1/T) * U - C
At temperature 4t, the entropy becomes:
s' =
- Entropy of a system of n particles at temperature t is given by s = a(nvu)^(1/3)
- u and v are the internal energy and volume of the system, respectively
- a is a constant
To find:
- Internal energy of the system when the temperature changes to 4t at constant volume
Solution:
Step 1: Relationship between Entropy and Temperature
The entropy of a system is related to the temperature through the equation:
ΔS = ∫(dQ/T)
Step 2: Entropy Change at Constant Volume
At constant volume, the heat transfer can be written as:
dQ = dU (change in internal energy)
Substituting this in the entropy equation:
ΔS = ∫(dU/T)
Step 3: Calculating the Change in Internal Energy
Using the given expression for entropy:
s = a(nvu)^(1/3)
Taking the derivative of s with respect to u:
ds/du = a(1/3)(nvu)^(-2/3) * nv
Rearranging the equation:
du = (3/(anv)) * (nvu)^(2/3) * ds
Step 4: Calculating the Internal Energy
Substituting the above expression for du in the entropy equation:
ΔS = ∫(dU/T)
ΔS = ∫(3/(anv)) * (nvu)^(2/3) * ds / T
Rearranging the equation:
ΔS = (3/(anvT)) * ∫(nvu)^(2/3) * ds
Integrating both sides:
ΔS = (3/(anvT)) * (3/5) * (nvu)^(5/3) + C
Where C is the constant of integration.
Step 5: Finding the Internal Energy
At constant volume, the change in entropy is given by:
ΔS = ∫(dU/T)
Substituting the expression for entropy:
(3/(anvT)) * (3/5) * (nvu)^(5/3) + C = ∫(dU/T)
Since the volume is constant, dv = 0. Therefore, the integral simplifies to:
(3/(anvT)) * (3/5) * (nvu)^(5/3) + C = ∫(dU/T) = (1/T) * ∫dU
Integrating both sides:
(3/(anvT)) * (3/5) * (nvu)^(5/3) + C = (1/T) * U
Simplifying the equation:
(3/(anvT)) * (3/5) * (nvu)^(5/3) = (1/T) * U - C
Since entropy is given by s = a(nvu)^(1/3), we can substitute this expression:
(3/(anvT)) * (3/5) * s^(5/3) = (1/T) * U - C
At temperature 4t, the entropy becomes:
s' =
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The entropy sof a system of n particles at a temperature t is given by s=a(nvu)^(1/3) where u and v are internal energy and volume of the system respectively and a is constant.if temperature changes to 4t at constant volume then internal energy of the system becomes (a) four times (b) two times (c) eight times (d) half?
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The entropy sof a system of n particles at a temperature t is given by s=a(nvu)^(1/3) where u and v are internal energy and volume of the system respectively and a is constant.if temperature changes to 4t at constant volume then internal energy of the system becomes (a) four times (b) two times (c) eight times (d) half? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about The entropy sof a system of n particles at a temperature t is given by s=a(nvu)^(1/3) where u and v are internal energy and volume of the system respectively and a is constant.if temperature changes to 4t at constant volume then internal energy of the system becomes (a) four times (b) two times (c) eight times (d) half? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The entropy sof a system of n particles at a temperature t is given by s=a(nvu)^(1/3) where u and v are internal energy and volume of the system respectively and a is constant.if temperature changes to 4t at constant volume then internal energy of the system becomes (a) four times (b) two times (c) eight times (d) half?.
The entropy sof a system of n particles at a temperature t is given by s=a(nvu)^(1/3) where u and v are internal energy and volume of the system respectively and a is constant.if temperature changes to 4t at constant volume then internal energy of the system becomes (a) four times (b) two times (c) eight times (d) half? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about The entropy sof a system of n particles at a temperature t is given by s=a(nvu)^(1/3) where u and v are internal energy and volume of the system respectively and a is constant.if temperature changes to 4t at constant volume then internal energy of the system becomes (a) four times (b) two times (c) eight times (d) half? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The entropy sof a system of n particles at a temperature t is given by s=a(nvu)^(1/3) where u and v are internal energy and volume of the system respectively and a is constant.if temperature changes to 4t at constant volume then internal energy of the system becomes (a) four times (b) two times (c) eight times (d) half?.
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