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A diatomic ideal gas is compressed adiabatically to 1/32 of its initial volume. If the initial temperature of the gas is Ti (in Kelvin) and the final temperature is aTi , the value of a is
Correct answer is '4'. Can you explain this answer?
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A diatomic ideal gas is compressed adiabatically to 1/32 of its initi...
Given:
- Diatomic ideal gas
- Compressed adiabatically to 1/32 of its initial volume
- Initial temperature: Ti (in Kelvin)
- Final temperature: aTi
To find:
The value of 'a'
Solution:
Adiabatic Process:
An adiabatic process is a thermodynamic process in which there is no exchange of heat between the system and its surroundings. In this process, the change in temperature and volume of the gas is related by the adiabatic equation:
P₁V₁^γ = P₂V₂^γ
Where:
- P₁ and P₂ are the initial and final pressures of the gas
- V₁ and V₂ are the initial and final volumes of the gas
- γ is the adiabatic index, which depends on the nature of the gas
Diatomic Ideal Gas:
A diatomic ideal gas consists of two atoms per molecule, such as oxygen (O₂) or nitrogen (N₂). For a diatomic ideal gas, the adiabatic index γ is equal to 7/5 or 1.4.
Applying the Adiabatic Equation:
In this problem, the gas is compressed adiabatically to 1/32 of its initial volume. Let the initial volume be V₁ and the final volume be V₂.
According to the adiabatic equation:
P₁V₁^γ = P₂V₂^γ
Since the initial and final pressures are not given, we can assume them to be equal.
P₁V₁^γ = P₂V₂^γ
P₁V₁^(7/5) = P₂(V₁/32)^(7/5)
P₁V₁^(7/5) = P₂(V₁^(7/5))/(32^(7/5))
P₁ = P₂/(32^(7/5))
Temperature Relationship:
In an adiabatic process, the relationship between temperature and volume is given by:
T₁V₁^(γ-1) = T₂V₂^(γ-1)
Substituting the values:
TiV₁^(7/5-1) = (aTi)V₂^(7/5-1)
TiV₁^(2/5) = aTiV₂^(2/5)
V₂/V₁ = (Ti/aTi)^(5/2)
V₂/V₁ = 1/(a^(5/2))
Since the initial volume is divided by 32, we have:
V₂/V₁ = 1/32
1/(a^(5/2)) = 1/32
a^(5/2) = 32
a = (32)^(2/5)
a = 2^(10/5)
a = 2^2
a = 4
Therefore, the value of 'a' is 4.
- Diatomic ideal gas
- Compressed adiabatically to 1/32 of its initial volume
- Initial temperature: Ti (in Kelvin)
- Final temperature: aTi
To find:
The value of 'a'
Solution:
Adiabatic Process:
An adiabatic process is a thermodynamic process in which there is no exchange of heat between the system and its surroundings. In this process, the change in temperature and volume of the gas is related by the adiabatic equation:
P₁V₁^γ = P₂V₂^γ
Where:
- P₁ and P₂ are the initial and final pressures of the gas
- V₁ and V₂ are the initial and final volumes of the gas
- γ is the adiabatic index, which depends on the nature of the gas
Diatomic Ideal Gas:
A diatomic ideal gas consists of two atoms per molecule, such as oxygen (O₂) or nitrogen (N₂). For a diatomic ideal gas, the adiabatic index γ is equal to 7/5 or 1.4.
Applying the Adiabatic Equation:
In this problem, the gas is compressed adiabatically to 1/32 of its initial volume. Let the initial volume be V₁ and the final volume be V₂.
According to the adiabatic equation:
P₁V₁^γ = P₂V₂^γ
Since the initial and final pressures are not given, we can assume them to be equal.
P₁V₁^γ = P₂V₂^γ
P₁V₁^(7/5) = P₂(V₁/32)^(7/5)
P₁V₁^(7/5) = P₂(V₁^(7/5))/(32^(7/5))
P₁ = P₂/(32^(7/5))
Temperature Relationship:
In an adiabatic process, the relationship between temperature and volume is given by:
T₁V₁^(γ-1) = T₂V₂^(γ-1)
Substituting the values:
TiV₁^(7/5-1) = (aTi)V₂^(7/5-1)
TiV₁^(2/5) = aTiV₂^(2/5)
V₂/V₁ = (Ti/aTi)^(5/2)
V₂/V₁ = 1/(a^(5/2))
Since the initial volume is divided by 32, we have:
V₂/V₁ = 1/32
1/(a^(5/2)) = 1/32
a^(5/2) = 32
a = (32)^(2/5)
a = 2^(10/5)
a = 2^2
a = 4
Therefore, the value of 'a' is 4.
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Community Answer
A diatomic ideal gas is compressed adiabatically to 1/32 of its initi...
For an adiabatic process, 
= constant
= constantTi
Substituting the given values, we get
For diatomic gas, y = 7/5
a = 327/5 - 1 = 322/5 = 22 = 4
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A diatomic ideal gas is compressed adiabatically to 1/32 of its initial volume. If the initial temperature of the gas is Ti (in Kelvin) and the final temperature is aTi , the value of a isCorrect answer is '4'. Can you explain this answer?
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