Combined Gas Law, PM = dRT & P = CRT

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Gas Equation Derivatives

Mastering the Combined Gas Law, $PM = dRT$, and $P = CRT$.

By chemca Team • Updated Aug 2026

The Ideal Gas Equation ($PV = nRT$) is incredibly versatile. By rearranging its variables, we can derive three powerful formulas that allow us to bypass tedious mole calculations in competitive exams. These are the Combined Gas Law, the Density formula, and the Concentration formula.

1. The Combined Gas Law

Often, a fixed amount of gas undergoes a change where Pressure, Volume, and Temperature all change simultaneously. In this case, Boyle's or Charles's laws alone are insufficient.

Derivation:

Start with the Ideal Gas Equation: $PV = nRT$.

Since the amount of gas is fixed, $n$ is constant. $R$ is also a constant. Therefore, we can group the variables on one side:

$$\frac{PV}{T} = nR = \text{Constant}$$

Because $\frac{PV}{T}$ is always a constant for a fixed mass of gas, we can equate two different states (State 1 and State 2):

$$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$$

Exam Tip: This equation is the ultimate multi-tool. If temperature is constant, cross out $T_1$ and $T_2$ and it magically becomes Boyle's Law ($P_1V_1 = P_2V_2$). If pressure is constant, cross out $P_1$ and $P_2$, and it becomes Charles's Law! Remember to always use Kelvin for temperature.

2. The Density & Molar Mass Equation ($PM = dRT$)

How do we find the density of a gas if we only know its pressure and temperature? We derive a new equation by substituting moles ($n$) with mass and molar mass.

Derivation:

1. We know that moles ($n$) = $\frac{\text{Given Mass } (m)}{\text{Molar Mass } (M)}$. Substitute this into $PV = nRT$:

$$P V = \left(\frac{m}{M}\right) R T$$

2. Rearrange the equation to group mass ($m$) and volume ($V$) together. We know Density ($d$) = $\frac{m}{V}$.

$$P = \left(\frac{m}{V}\right) \frac{RT}{M} \quad \implies \quad P = \frac{d R T}{M}$$

3. Multiply both sides by Molar Mass ($M$) to get the final, memorable form:

$$P M = d R T$$

• $P$ = Pressure
• $M$ = Molar Mass (Molecular Weight)
• $d$ = Density of the gas
• $R$ = Universal Gas Constant
• $T$ = Temperature (in Kelvin)

3. The Concentration Equation ($P = CRT$)

In chemical kinetics, equilibrium ($K_p$ vs $K_c$), and solutions, we often deal with concentration rather than volume. We can easily adapt the gas equation for this.

Derivation:

Start with $PV = nRT$ and rearrange to solve for Pressure ($P$):

$$P = \left(\frac{n}{V}\right) R T$$

The term $\frac{n}{V}$ (moles per unit volume) is the exact definition of Molarity or Concentration ($C$).

$$P = C R T$$

Crucial Application: This exact formula ($P = CRT$ or $\pi = CRT$) is also used later in Class XII Solutions to calculate Osmotic Pressure ($\pi$). It shows that the pressure of an ideal gas is directly proportional to its concentration.

4. Key Proportionalities to Memorize

From Equation	Direct Proportionality	Inverse Proportionality
$PM = dRT$	$d \propto P$ $d \propto M$	$d \propto \frac{1}{T}$
$P = CRT$	$P \propto C$ $P \propto T$	$C \propto \frac{1}{T}$ (at constant $P$)

Knowledge Check

Test your ability to apply these derived equations

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