Change in internal energy due to adiabatic expansion of ideal gas

In summary, the change in internal energy of an ideal gas when it is expanded adiabatically from (Po,Vo) to (P,V) is ΔU = (CV/R)(PV - PoVo) = 1/(r-1)(PV - PoVo), where r = CP/Cv. This value does not depend on the process used to reach the final state, as internal energy is a state variable.
  • #1
sudipmaity
50
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what is the change In internal energy of an ideal gas when it is expanded Adiabatically from (p °,v°) to ( p,v )?
The relevant equations :
dQ= dU+ pdv ; pV^ r = K (constant). r = Cp / Cv.
Attempted solution :
During adiabatic process dQ=0 ; p= k v ^ -r
Th f r : du= -k v^ -rdv
Integrating the above relation : Uf-U i = -k [ ( v ^ -r + 1 ) / -r + 1 ] v to v° .
= -k [ {(v ^ 1-r ) / (1-r)} -{(v° ^ 1-r ) /( 1-r) } ]
= 1 / (1-r)[ (k v ^ -r) v-( k v° ^-r ) v° ] = 1/ (1-r) [ pv-p° v° ] .
IS there any mistake ? Am i absolutely right ?? .
 
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  • #2
sudipmaity said:
Integrating the above relation : Uf-U i = -k [ ( v ^ -r + 1 ) / -r + 1 ] v to v° .
= -k [ {(v ^ 1-r ) / (1-r)} -{(v° ^ 1-r ) /( 1-r) } ]
= 1 / (1-r)[ (k v ^ -r) v-( k v° ^-r ) v° ] = 1/ (1-r) [ pv-p° v° ] .
IS there any mistake ? Am i absolutely right ?? .

I believe you dropped the negative sign in front of the k when writing the last line. Otherwise, it looks good to me.

Using r = CP/Cv and CP-Cv = R , you can rewrite 1/(1-r) in terms of R and Cv if you wish.
 
  • #3
My bad .what I intended to write was 1/(r-1)( pv-p° v° )
 
  • #4
Good.

There's another approach to this problem. Internal energy U is a state variable. So, the change in U between the state (Po,Vo) and (P,V) doesn't depend on whether or not the process was adiabatic. Any process connecting those two states gives the same ΔU.

For an ideal gas, U = nCvT. Using the ideal gas law, PV = nRT, this can be written as U = (CV/R)PV. So, ΔU = (CV/R)(PV - PoVo) and you can show that CV/R = 1/(r-1).
 
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Likes gracy
  • #5


Your solution looks correct. The change in internal energy of an ideal gas during adiabatic expansion can be calculated using the formula Uf-Ui = 1/(1-r) [pv-pi v] where r is the ratio of specific heats (Cp/Cv). This formula is derived from the first law of thermodynamics, dQ = dU + pdV, where dQ=0 for adiabatic processes. So, your solution is accurate and you are on the right track! Keep up the good work.
 

1. What is adiabatic expansion?

Adiabatic expansion is a thermodynamic process in which a gas expands without any exchange of heat with its surroundings. This means that the internal energy of the gas remains constant during the expansion.

2. How is the change in internal energy of an ideal gas calculated during adiabatic expansion?

The change in internal energy (ΔU) of an ideal gas during adiabatic expansion can be calculated using the equation ΔU = -W, where W is the work done by the gas during expansion. This is based on the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system.

3. What factors affect the change in internal energy during adiabatic expansion of an ideal gas?

The change in internal energy during adiabatic expansion of an ideal gas is affected by the initial and final volumes of the gas, as well as the specific heat capacity ratio of the gas. It is also influenced by the type of process (reversible or irreversible) and the type of gas (monoatomic or diatomic).

4. How does the change in internal energy during adiabatic expansion differ from that during isothermal expansion?

In adiabatic expansion, the internal energy of the gas remains constant. In contrast, during isothermal expansion, the temperature of the gas remains constant, resulting in a change in internal energy. This is because in adiabatic expansion, there is no heat exchange, while in isothermal expansion, heat is added or removed to maintain a constant temperature.

5. Why is adiabatic expansion important in thermodynamics?

Adiabatic expansion is important in thermodynamics because it allows for the study and understanding of the behavior of gases under specific conditions, such as when there is no heat exchange. It also has practical applications, such as in the operation of heat engines and refrigeration systems.

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