Obtaining gamma (γ) in an adiabatic, reversible expansion

In summary, we need to calculate the final temperature T2 using the equation T2 = T1(P1/P2)((1-γ)/γ), where γ = 5/3 for a monatomic gas. Then, we can use the values of T1 and T2 to calculate q, w, ΔU, and ΔH using the equations q = 0, ΔU = w = CvΔT, and ΔH = CpΔT. The value of γ for a monatomic gas can be obtained by using the equation \gamma = \frac{C_P}{C_V} = \frac{f+2
  • #1
anisotropic
59
0
PROBLEM:

One mole of an ideal monatomic gas initially at 300 K (T1) and a pressure of 15.0 atm (P1) expands to a final pressure of 1.00 atm (P2). The expansion occurs via an adiabatic and reversible path. Calculate q, w, ΔU, and ΔH.


SOLUTION:

q = 0 (adiabatic; no heat exchange occurs)
Thus, ΔU = w = CvΔT

ΔH = CpΔT

ΔT = T2 - T1

Need T2 to calculate values (not an isothermal expansion)...

Known equation for reversible adiabatic process: P1V1γ = P2V2γ (γ = gamma)

Using PV = nRT, substitute in and simplify to obtain expression for T2...

T2 = T1(P1/P2)((1-γ)/γ)

*where problems arise...

γ = ?

The solutions manual says γ = 5/3, but I don't know how this value is obtained.

I do know that, as an ideal monatomic gas, the internal energy (U) of the gas (per mole?) is: 3/2RT (we assume only translations occur). But how is this related to Cv, Cp, and more importantly, γ?
 
Last edited:
Physics news on Phys.org
  • #2
γ can be obtained if you know the degrees of freedom for an ideal mono atomic gas.
 
  • #3
anisotropic said:
The solutions manual says γ = 5/3, but I don't know how this value is obtained.

I do know that, as an ideal monatomic gas, the internal energy (U) of the gas (per mole?) is: 3/2RT (we assume only translations occur). But how is this related to Cv, Cp, and more importantly, γ?


[itex]\gamma = \frac{C_P}{C_V} = \frac{f+2}{f}[/itex]

Where f is the number of degrees of freedom of the gas.

Taking the monoatomic case, the number of degrees of freedom are 3. So, [itex]\frac{2+3}{3} = \frac{5}{3}[/itex]

Note that the internal energy U of one mole monoatomic ideal gas is

[itex]\frac{3}{2}RT[/itex]

But in general, for any ideal gas, internal energy is

[itex]\frac{1}{2}fnRT[/itex]
 

1. What is gamma (γ) in the context of adiabatic, reversible expansion?

Gamma, also known as the adiabatic index or heat capacity ratio, is a dimensionless quantity that describes the relationship between the pressure and volume of a gas in an adiabatic process. It is defined as the ratio of the specific heat at constant pressure to the specific heat at constant volume.

2. How is gamma (γ) obtained in an adiabatic, reversible expansion?

Gamma can be obtained by measuring the change in pressure (ΔP) and volume (ΔV) of the gas during an adiabatic, reversible expansion, and using the equation γ = -ΔV/ΔP.

3. Why is adiabatic, reversible expansion important in thermodynamics?

Adiabatic, reversible expansion is important in thermodynamics because it is a fundamental process that is used to understand the behavior of gases. It allows us to study the relationship between pressure, volume, and temperature in a controlled environment, and is essential in the development of thermodynamic principles and laws.

4. How does gamma (γ) affect the efficiency of an engine?

The value of gamma (γ) can affect the efficiency of an engine, as it is related to the amount of work that can be extracted from a gas during adiabatic expansion. A higher value of gamma means that more work can be obtained, resulting in a more efficient engine.

5. Can gamma (γ) change during an adiabatic, reversible expansion?

No, gamma (γ) remains constant during an adiabatic, reversible expansion as it is a characteristic of the gas itself and does not depend on external factors. It can only change if there is a change in the composition of the gas or if the gas undergoes a chemical reaction.

Similar threads

  • Introductory Physics Homework Help
Replies
1
Views
791
  • Biology and Chemistry Homework Help
Replies
4
Views
2K
Replies
22
Views
2K
  • Introductory Physics Homework Help
Replies
8
Views
871
  • Biology and Chemistry Homework Help
Replies
2
Views
1K
Replies
56
Views
3K
  • Biology and Chemistry Homework Help
Replies
11
Views
3K
  • Introductory Physics Homework Help
Replies
8
Views
664
  • Engineering and Comp Sci Homework Help
Replies
3
Views
3K
  • Biology and Chemistry Homework Help
Replies
1
Views
3K
Back
Top