Formula Used
At constant volume ($V = \text{constant}$), work done by the system is zero ($W = 0$). According to the first law of thermodynamics:
- $\Delta U$ = Change in internal energy
- $n$ = Number of moles (or mass $m$)
- $C_v$ = Heat capacity at constant volume
- $\Delta T$ = Change in temperature ($T_2 - T_1$)
Calculator
How to Use
- Select whether you are inputting moles or mass for your substance quantity.
- Enter the numeric value of the amount in the designated input field.
- Input the specific or molar heat capacity value ($C_v$) at constant volume.
- Provide both initial and final temperatures along with the proper unit.
- Click the calculate button to see instant internal energy results.
Understanding Internal Energy Change at Constant Volume
Thermodynamics is a core pillar of physics and chemistry, studying how heat, work, and energy interact within physical systems. When analyzing closed systems, one of the most critical scenarios involves processes occurring at a constant volume, technically known as isoric or isochoric processes. In such conditions, the physical boundaries of the system are completely rigid, meaning the system cannot expand or contract. Consequently, the boundary mechanical work done by or on the system is identically zero ($W = 0$).
According to the First Law of Thermodynamics, energy is conserved across all processes. The mathematical statement of this law is expressed as $\Delta U = Q - W$. Because the volume remains completely fixed during an isochoric process, no expansion work takes place. Substituting $W = 0$ directly simplifies the equation to $\Delta U = Q$. This fundamental relationship indicates that any heat added to or removed from the system at constant volume goes entirely into changing the internal kinetic and potential energy of the constituent particles, directly reflecting as a shift in system temperature.
The Role of Heat Capacity
To quantify this change accurately, physicists utilize the molar heat capacity at constant volume, denoted as $C_v$. Heat capacity measures how much thermal energy is required to raise the temperature of a given quantity of a substance by one unit of temperature. Because internal energy for an ideal gas depends exclusively on temperature, the expression scales linearly with temperature difference. Multiplying the amount of substance ($n$ or $m$), the constant-volume heat capacity ($C_v$), and the temperature difference ($\Delta T$) yields the precise change in internal energy.