Calculate Change in Internal Energy Given Dp Dv N

Introduction

The change in internal energy (ΔU) is a key quantity in thermodynamics that describes how the energy content of a system changes as it undergoes a process. For many processes, especially those involving ideal gases, the change in internal energy can be calculated using the first law of thermodynamics, which relates the change in internal energy to heat added to the system and work done by the system.

When dealing with processes that involve changes in pressure and volume, the change in internal energy can be expressed in terms of these variables. This is particularly useful for processes where the heat capacity at constant volume (Cv) is known or can be approximated.

Formula

The change in internal energy (ΔU) for a process involving changes in pressure (ΔP) and volume (ΔV) can be calculated using the following formula:

Where:

• ΔU is the change in internal energy (in joules, J)
• n is the number of moles of the gas (in moles, mol)
• Cv is the molar heat capacity at constant volume (in joules per mole per kelvin, J/mol·K)
• ΔT is the change in temperature (in kelvin, K)
• P is the pressure (in pascals, Pa)
• ΔV is the change in volume (in cubic meters, m³)

This formula combines the contributions to the change in internal energy from the temperature change (nCvΔT) and the work done by the system (PΔV).

Calculation

To calculate the change in internal energy, you need to know the values of n, Cv, ΔT, P, and ΔV. The calculator on the right provides a convenient way to input these values and compute the result.

The calculation involves the following steps:

1. Input the number of moles (n) of the gas.
2. Input the molar heat capacity at constant volume (Cv).
3. Input the change in temperature (ΔT).
4. Input the pressure (P).
5. Input the change in volume (ΔV).
6. Click the "Calculate" button to compute the change in internal energy (ΔU).

The result will be displayed in joules (J), which is the standard unit for energy in the International System of Units (SI).

Example

Let's consider an example to illustrate how to use the calculator. Suppose we have 2 moles of an ideal gas with a molar heat capacity at constant volume of 20 J/mol·K. The gas undergoes a process where the temperature increases by 10 K, the pressure is 100,000 Pa, and the volume increases by 0.01 m³.

In this example, the change in internal energy is 1,400 joules. This result can be verified using the calculator by entering the given values and clicking the "Calculate" button.