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Consider a photon gas inside a 3-D cavity of volume V and in thermal equilibrium at temperature T. If photon gas is expanded adiabatically into vacuum to 16 times of initial volume, then its temperaure changes to n times of initial tempeature. The value of n is _________________.
Correct answer is '0.5'. Can you explain this answer?
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Consider a photon gas inside a 3-D cavity of volume V and in thermal e...
The process is free expansion For free expansion internal energy (U) = constant.
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Consider a photon gas inside a 3-D cavity of volume V and in thermal e...
Answer:
Given data:
- Photon gas in a 3-D cavity of volume V at temperature T.
- Adiabatic expansion of photon gas to 16 times of initial volume.
- Temperature changes to n times of initial temperature.
We need to find the value of n.
Solution:
Adiabatic expansion of photon gas means that there is no exchange of heat between the photon gas and the surroundings during the expansion process. Therefore, the initial energy of the photon gas should be conserved.
Let the initial number of photons in the cavity be N and their energy density be u. Therefore, the initial energy of the photon gas can be expressed as E = N * u * V.
The final volume of the cavity is 16 times the initial volume. Therefore, the final energy density of the photon gas can be expressed as u' = u/16.
Since the initial energy of the photon gas is conserved, we can write:
E = N * u * V = N * u' * (16V)
Thus, we get:
u' = u/16 = E/(16NV)
We know that the energy density of a photon gas is given by:
u = (π^2/15) * (kT)^4 / (ħ^3 * c^3)
where k is the Boltzmann constant, ħ is the reduced Planck constant, and c is the speed of light.
Therefore, we can express u' as:
u' = (π^2/15) * (kT')^4 / (ħ^3 * c^3)
where T' is the final temperature of the photon gas.
Comparing the expressions for u' obtained above, we get:
(kT')^4 = 16(kT)^4
Therefore, we get:
T' = (16)^(1/4) * T
Thus, the value of n, i.e., the ratio of the final temperature to the initial temperature, is:
n = T' / T = (16)^(1/4)
n = 2^(1/2) = 0.5
Therefore, the value of n is 0.5.
Conclusion:
- The initial energy of the photon gas is conserved during adiabatic expansion.
- The final energy density of the photon gas is u' = u/16.
- The final temperature of the photon gas is T' = (16)^(1/4) * T.
- The value of n, i.e., the ratio of the final temperature to the initial temperature, is 0.5.
Given data:
- Photon gas in a 3-D cavity of volume V at temperature T.
- Adiabatic expansion of photon gas to 16 times of initial volume.
- Temperature changes to n times of initial temperature.
We need to find the value of n.
Solution:
Adiabatic expansion of photon gas means that there is no exchange of heat between the photon gas and the surroundings during the expansion process. Therefore, the initial energy of the photon gas should be conserved.
Let the initial number of photons in the cavity be N and their energy density be u. Therefore, the initial energy of the photon gas can be expressed as E = N * u * V.
The final volume of the cavity is 16 times the initial volume. Therefore, the final energy density of the photon gas can be expressed as u' = u/16.
Since the initial energy of the photon gas is conserved, we can write:
E = N * u * V = N * u' * (16V)
Thus, we get:
u' = u/16 = E/(16NV)
We know that the energy density of a photon gas is given by:
u = (π^2/15) * (kT)^4 / (ħ^3 * c^3)
where k is the Boltzmann constant, ħ is the reduced Planck constant, and c is the speed of light.
Therefore, we can express u' as:
u' = (π^2/15) * (kT')^4 / (ħ^3 * c^3)
where T' is the final temperature of the photon gas.
Comparing the expressions for u' obtained above, we get:
(kT')^4 = 16(kT)^4
Therefore, we get:
T' = (16)^(1/4) * T
Thus, the value of n, i.e., the ratio of the final temperature to the initial temperature, is:
n = T' / T = (16)^(1/4)
n = 2^(1/2) = 0.5
Therefore, the value of n is 0.5.
Conclusion:
- The initial energy of the photon gas is conserved during adiabatic expansion.
- The final energy density of the photon gas is u' = u/16.
- The final temperature of the photon gas is T' = (16)^(1/4) * T.
- The value of n, i.e., the ratio of the final temperature to the initial temperature, is 0.5.
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Consider a photon gas inside a 3-D cavity of volume V and in thermal equilibrium at temperature T. If photon gas is expanded adiabatically intovacuum to 16 times of initial volume, then its temperaure changes to n times of initial tempeature. The value of n is _________________.Correct answer is '0.5'. Can you explain this answer?
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Consider a photon gas inside a 3-D cavity of volume V and in thermal equilibrium at temperature T. If photon gas is expanded adiabatically intovacuum to 16 times of initial volume, then its temperaure changes to n times of initial tempeature. The value of n is _________________.Correct answer is '0.5'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider a photon gas inside a 3-D cavity of volume V and in thermal equilibrium at temperature T. If photon gas is expanded adiabatically intovacuum to 16 times of initial volume, then its temperaure changes to n times of initial tempeature. The value of n is _________________.Correct answer is '0.5'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a photon gas inside a 3-D cavity of volume V and in thermal equilibrium at temperature T. If photon gas is expanded adiabatically intovacuum to 16 times of initial volume, then its temperaure changes to n times of initial tempeature. The value of n is _________________.Correct answer is '0.5'. Can you explain this answer?.
Consider a photon gas inside a 3-D cavity of volume V and in thermal equilibrium at temperature T. If photon gas is expanded adiabatically intovacuum to 16 times of initial volume, then its temperaure changes to n times of initial tempeature. The value of n is _________________.Correct answer is '0.5'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider a photon gas inside a 3-D cavity of volume V and in thermal equilibrium at temperature T. If photon gas is expanded adiabatically intovacuum to 16 times of initial volume, then its temperaure changes to n times of initial tempeature. The value of n is _________________.Correct answer is '0.5'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a photon gas inside a 3-D cavity of volume V and in thermal equilibrium at temperature T. If photon gas is expanded adiabatically intovacuum to 16 times of initial volume, then its temperaure changes to n times of initial tempeature. The value of n is _________________.Correct answer is '0.5'. Can you explain this answer?.
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Consider a photon gas inside a 3-D cavity of volume V and in thermal equilibrium at temperature T. If photon gas is expanded adiabatically intovacuum to 16 times of initial volume, then its temperaure changes to n times of initial tempeature. The value of n is _________________.Correct answer is '0.5'. Can you explain this answer?, a detailed solution for Consider a photon gas inside a 3-D cavity of volume V and in thermal equilibrium at temperature T. If photon gas is expanded adiabatically intovacuum to 16 times of initial volume, then its temperaure changes to n times of initial tempeature. The value of n is _________________.Correct answer is '0.5'. Can you explain this answer? has been provided alongside types of Consider a photon gas inside a 3-D cavity of volume V and in thermal equilibrium at temperature T. If photon gas is expanded adiabatically intovacuum to 16 times of initial volume, then its temperaure changes to n times of initial tempeature. The value of n is _________________.Correct answer is '0.5'. Can you explain this answer? theory, EduRev gives you an
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