Question

Which equation is valid for adiabatic process?

• A. \(TV^{\gamma-1} = \text{constant}\)
• B. \(pV^{\gamma} = \text{constant}\)
• C. \(T^{\gamma}V^{\gamma-1} = \text{constant}\)
• D. \(\frac{p^{1-\gamma}}{T^{\gamma}} = \text{constant}\)

Hint:

Always remember two key adiabatic forms: \(pV^\gamma\) and \(TV^{\gamma-1}\).

Answer 1

The Correct Option is A

In thermodynamics, an adiabatic process is one in which no heat is transferred to or from the system. For an ideal gas undergoing an adiabatic process, there are specific equations that relate the pressure \(p\), volume \(V\), and temperature \(T\). One of these equations is:

• \(TV^{\gamma-1} = \text{constant}\)

Here, \(\gamma\) (gamma) is the heat capacity ratio, also known as the adiabatic index. This equation indicates that, for an adiabatic process, the product of temperature \(T\) and volume \(V\) raised to the power \(\gamma - 1\) remains constant.

Let's verify why this equation is appropriate for an adiabatic process and rule out the other options:

1. \(TV^{\gamma-1} = \text{constant}\): This is a valid equation for an adiabatic process as explained above.
2. \(pV^{\gamma} = \text{constant}\): This is another valid form for an adiabatic process equation, but it is not given as the correct answer in this instance.
3. \(T^{\gamma}V^{\gamma-1} = \text{constant}\): This equation is not a standard form for describing adiabatic processes and is incorrect.
4. \(\frac{p^{1-\gamma}}{T^{\gamma}} = \text{constant}\): This is not a valid equation for describing an adiabatic process either.

Hence, the correct equation for the adiabatic process given in this question is \(TV^{\gamma-1} = \text{constant}\). This is consistent with the laws of thermodynamics governing adiabatic processes.