What`s the formulas for heat,work done and internal energy in an polytropic process

In a polytropic process, which is a thermodynamic process that follows the relation \( PV^n = \text{constant} \), the formulas for heat, work done, and internal energy can be derived from the principles of thermodynamics. Let’s break down each of these concepts step by step.

Understanding the Polytropic Process

A polytropic process is characterized by a specific heat capacity ratio, denoted as \( n \). This ratio can take various values, leading to different types of processes, such as isothermal (n=1), adiabatic (n=γ, where γ is the ratio of specific heats), and isobaric (n=0). The general equation for a polytropic process is:

PV^n = C

Work Done in a Polytropic Process

The work done \( W \) during a polytropic process can be calculated using the formula:

W = \frac{P_2 V_2 - P_1 V_1}{1 - n} (for \( n \neq 1 \))

Where:

• \( P_1, V_1 \) are the initial pressure and volume.
• \( P_2, V_2 \) are the final pressure and volume.

If \( n = 1 \), the process is isothermal, and the work done can be calculated using:

W = nRT \ln\left(\frac{V_2}{V_1}\right)

Heat Transfer in a Polytropic Process

The heat transfer \( Q \) in a polytropic process can be expressed as:

Q = ΔU + W

Where \( ΔU \) is the change in internal energy. For an ideal gas, the change in internal energy can be calculated as:

ΔU = nC_v(T_2 - T_1)

Here, \( C_v \) is the specific heat at constant volume, and \( T_1, T_2 \) are the initial and final temperatures, respectively.

Internal Energy Change

The change in internal energy \( ΔU \) for an ideal gas during a polytropic process can also be expressed as:

ΔU = \frac{R}{n-1}(T_2 - T_1)

This formula is derived from the first law of thermodynamics and the ideal gas law, where \( R \) is the universal gas constant.

Putting It All Together

To summarize, in a polytropic process, you can calculate:

• The work done using the formula \( W = \frac{P_2 V_2 - P_1 V_1}{1 - n} \) for \( n \neq 1 \).
• The heat transfer using \( Q = ΔU + W \), where \( ΔU \) can be calculated based on the specific heat at constant volume.
• The change in internal energy using \( ΔU = nC_v(T_2 - T_1) \) or \( ΔU = \frac{R}{n-1}(T_2 - T_1) \).

These relationships are fundamental in understanding how energy is transferred and transformed in thermodynamic systems, particularly in processes that do not fit neatly into the categories of isothermal or adiabatic processes.