Work done by gas in adiabatic process

[collectedsoul]

In an adiabatic process PV^γ=constant

Now I thought the work done by an ideal gas in an adiabatic process was given by the equation here: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html

But while doing a GRE question the answer was given as P_fV_f - P_iV_i / 1 - γ

Is this correct (it must be!)? And if so why haven't I seen this equation anywhere before?

 

[CAF123]

I think the two are equivalent. Reexpress the formula in the link using, for an adiabatic process, $$P_1 V_1^{\gamma} = P_2 V_2^{\gamma}$$

 

[Andrew Mason]

In an adiabatic process PV^γ=constant

Now I thought the work done by an ideal gas in an adiabatic process was given by the equation here: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html

But while doing a GRE question the answer was given as P_fV_f - P_iV_i / 1 - γ

Is this correct (it must be!)? And if so why haven't I seen this equation anywhere before?

Did you mean that the GRE answer was W = (P_fV_f - P_iV_i) / (1 - γ) ?

Since it is a reversible adiabatic process (Q=0), the work done BY the gas has to be equal and opposite to its change in internal energy: W = - ΔU (first law).

For an ideal gas, ΔU = nCvΔT = nCv(γ-1)/(γ-1)ΔT = nRΔT/(γ-1) = Δ(PV)/(γ-1) = (PfVf - PiVi)/(γ-1).

AM

 

[CAF123]

Alternatively, $$P_i V_i^{\gamma} = P_f V_f^{\gamma} = K \Rightarrow KV_i^{1-\gamma} = P_iV_i\,\,\text{and}\,\,KV_f^{1-\gamma} = P_fV_f$$
Then sub in.

 

[PJC01]

Yes, the equation given in the GRE question is correct. The work done by an ideal gas in an adiabatic process can be calculated using the equation W = (PfVf - PiVi) / (1 - γ), where Pf and Vf are the final pressure and volume of the gas, Pi and Vi are the initial pressure and volume of the gas, and γ is the adiabatic index.

This equation is derived from 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. In an adiabatic process, there is no heat exchange between the gas and its surroundings, so the change in internal energy is equal to the work done by the gas.

The equation you have seen before, PVγ=constant, is known as the adiabatic equation of state. It is derived from the ideal gas law and is used to describe the relationship between pressure and volume in an adiabatic process. However, this equation does not directly give the work done by the gas.

The equation given in the GRE question is a more specific and accurate representation of the work done by an ideal gas in an adiabatic process. It takes into account the initial and final states of the gas, as well as the adiabatic index, which is a measure of the gas's ability to transfer heat. This equation is commonly used in thermodynamics and can be found in many textbooks and online resources.