# Combined Gas Law Calculator

Created by Davide Borchia
Last updated: Aug 24, 2022

Compress, heat, pump, or do all this simultaneously: our combined gas law calculator will help you understand the math behind thermodynamical processes involving ideal gases.

In this exhaustive article, you will learn:

• What are thermodynamical processes: isothermal, isobaric, and isochoric processes (plus the adiabatic process);
• The thermodynamics of the processes: calculate the internal energy variation, the absorbed heat, and the work done;
• In-detail explanation of each process; and
• A quick primer on how to use our calculator.

## What is an ideal gas? What are thermodynamic processes?

An ideal gas is a gas (a state of matter that identifies compressible fluids) in which intermolecular forces are negligible. The ideal gas law describes the behavior of such gas and, in real-world situations, the behavior of low-density gases.

Three state variables define a gas:

• $p$ — The pressure;
• $T$ — The temperature; and
• $V$ — The volume.

We can add the number of moles, $n$, to the set.

If one of those three state variables remains constant durant a transformation, the other two will change in particular ways. Such transformations follow rather straightforward rules and a rigid underlying physics: these facts awarded them the distinctive name of thermodynamic processes.

Scientists discovered those processes well before truly understanding the behavior of gases. It's time to introduce the combined gas law equation and the type of thermodynamical processes.

## The combined gas law equation: four thermodynamic process in one

Let's list the three possible transformations we identify when we "lock" one of the state variables:

• If the temperature remains constant, we are dealing with an isothermal process: the equation for such transformation comes from Boyle's law.
• If it's the volume to remain constant, we are talking of isochoric processes. Gay-Lussac's law describes them.
• When the pressure doesn't change during the transformation, we have an isobaric process, of which Charles' law provides a simple mathematical description.

In addition to these three processes, we can identify one last type of transformation: adiabatic transformation. In such a process, none of the state variables remain constant: it is somehow special since it doesn't involve a heat transfer to the environment.

We can describe all these processes with a single law, the ideal gas law (check our ideal gas law calculator to learn more about it). The unmistakable formula for every science student is:

$P\cdot V = n\cdot R \cdot T$

We only need to introduce $R$, the ideal gas constant:

$R = 8.314462618\ \frac{\text{J}}{\text{K}\cdot\text{mol}}$

The combined gas law is almost identical to the ideal gas law: we only ignore the effect of the number of moles, absorbing it in the constant (for each process) $k = n\cdot R$:

$p\cdot V = k\cdot T$

Or, in an even better shape:

$\frac{p\cdot V}{T} = k$

Before calculating the combined gas law in every possible situation, we must think about thermodynamics again.

## Calculate internal energy, absorbed heat, and work

Speaking of heat, we can introduce another functionality of our combined gas law calculator. During the thermodynamic processes, not only do the state variables change, but the other three quantities vary in specific ways. We are talking of:

• $\Delta U$ — The variation in internal energy;
• $Q$ — The absorbed heat; and
• $W$ — The work performed on the environment by the gas.

To calculate these quantities, we often need to know the heat capacity of the gas. For ideal gases, there is an easy way to do so: counting the atoms.

The molar heat capacity for constant volume, $C_{\text{V}}$ is:

• $C_{\text{V}} = \tfrac{3}{2}R$ — For monoatomic gases ($\text{Ne}$, $\text{Ar}$, etc.).
• $C_{\text{V}} = \tfrac{5}{2}R$ — For diatomic gases ($\text{O}_2$, $\text{H}_2$, etc.).
• $C_{\text{V}} = 3\cdot R$ — For every other polyatomic gas.

The molar heat capacity for constant pressure, $C_{\text{p}}$ is closely related to $C_{\text{V}}$:

$C_{\text{p}}=C_{\text{V}}+R$

To calculate the internal energy variation, we apply the following formula:

$\Delta U = C_{\text{V}}\cdot n\cdot \Delta T$

Where $\Delta T$ is the temperature variation in the thermodynamic process.

We can equate the internal energy variation to the other two quantities in the following way:

$\Delta U = Q - W$

To calculate the work, we need to consider the integral of the pressure with respect to the volume:

$W = \int_{V_1}^{V_2} p(V)\text{d}V$

Worry not: it won't be hard to solve in most cases!

Now you have all the fundamental tools to study the thermodynamic processes: let's analyze them one by one.

## Isothermal processes: equation and explanation

Isothermal processes happen when the temperature of the gas doesn't change:

$T_1=T_2$

Boyle's law tells us that, in these conditions, pressure and volume are inversely proportional. An isothermal process has equation:

$p_1\cdot V_1 = p_2 \cdot V_2$

In such a process, the internal energy doesn't change: we can equate work and absorbed heat:

$Q=W$

The dependency of the pressure on the volume is, rather straightforwardly, $P = A/V$, where $A$ is a constant. The result of the integration is:

$Q=W = \frac{A}{\ln{(\frac{V_2}{V_1})}} = \frac{n\cdot R\cdot T}{\ln{(\frac{V_2}{V_1})}}$

🙋 To realize an isothermal process in a laboratory, you must slowly change pressure and/or volume.

## Isobaric process: formula, interpretation, and examples

If we keep the pressure constant during a process, we realize an isobaric process (from Ancient Greek isos, same, and baros, weight). To achieve this condition in a real-life situation, leave the lid of your vessel open!

In an isobaric process, a formula relates temperature and volume according to Charles' law:

$\frac{V_1}{T_1} = \frac{V_2}{T_2}$

Notice how, in this case, the two quantities are directly proportional.

In an isobaric process, the formula for the work is simplified by the fact that the pressure remains constant:

$W =\int p\text{d}V=p\cdot \Delta V$

To calculate the absorbed heat, we use the variation in internal energy associated with the (known) variation in temperature:

\begin{align*} Q \!&=\! \Delta U\!+\! W \!=\! C_{\text{V}}\!\cdot \!n\! \cdot \!\Delta T\! +\! p \!\cdot \!\Delta V\\ &\!=\!C_{\text{V}}\!\cdot \!n\!\cdot \!\Delta T \!+\! n\!\cdot \!R\!\cdot\!\Delta T\\ &=C_{\text{p}}\!n\!\cdot \!\Delta T \end{align*}

Where we used:

• The relationship between internal energy, work, and heat;
• The ideal gas law;
• The relationship between heat capacity under constant volume and pressure.

## Formulas for isochoric processes: Gay-Lussac's law and thermodynamics

When you keep your vessel closed, the volume can't change: you set up the perfect ground for an isochoric process: the formula that describes such change shows, once again, direct proportionality between the quantities involved: temperature and pressure:

$\frac{p_1}{T_1} = \frac{p_2}{T_2}$

Which is nothing but Gay-Lussac's law.

In an isochoric process, the work exerted by the gas is, of course, null: the variation in volume is zero:

$W = 0$

The internal energy and the temperature are, then, identical:

$\Delta U = Q = C_{\text{V}}\cdot n \cdot \Delta T$

## Adiabatic processes: formulas for adiabatic compression and expansion

To complete our combined gas law calculator, we need to introduce the adiabatic processes. As said before, an adiabatic process is a thermodynamic process where pressure, temperature, and volume all change, but there is no heat exchange with the environment. The formula for an adiabatic compression (or expansion) is:

$p_1 \cdot V_1^{\gamma} = p_2 \cdot V_2^{\gamma}$

$\gamma$ is an adimensional parameter equal to the ratio between the two heat capacities: $\gamma = C_{\text{p}}/C_{\text{V}}$.

The work done or to which the gas is subjected is entirely converted into a variation in the internal energy:

$\Delta U = - W$

How do you achieve such conditions? Press (or pull) the gas fast enough: it won't have time to exchange heat.

## How to use our combined gas law calculator

Now that you know all four thermodynamical processes, it's time to explain how to use our combined gas law calculator.

You can calculate one process at a time. Select the one you need from the first variable, process type. Now choose the gas: we need to know this to calculate the variation in internal energy.

Note that you won't have to search for molar mass and heat capacities: we already know them!

Next, give us the initial parameters: if you insert pressure, temperature, and volume, we'll calculate the number of moles. Otherwise, you may have to add that value manually. These calculations come from the ideal gas law.

Final step: choose where to stop. Insert one (or more) final values. We will fill the desired field if you give us a valid combination. If you give us enough data, our calculator will return you many values: be sure to understand them all!

Davide Borchia
Select process and gas type
Process
Working gas
Nitrogen
Insert initial parameters
Initial pressure (p₁)
Pa
Initial volume (V₁)
Initial temperature (T₁)
K
Resulting parameters
Final pressure (p₂)
Pa
Final volume (V₂)
Final temperature (T₂)
K
Internal energy change (ΔU)
J
Work (W)
J
Heat (Q)
J
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