The Ideal Gas Law and Combined Gas Law: Understanding and Applications
The Ideal Gas Law: A Universal Equation for Gases
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas which relates the pressure, volume, temperature, and amount of substance in a gas.
The ideal gas law is a mathematical relationship that connects four key variables describing a gas: pressure ($P$), volume ($V$), temperature ($T$), and the number of moles ($n$):
$$PV = nRT$$
Here:
- $P$: Pressure (measured in pascals, Pa),
- $V$: Volume (measured in cubic meters, m³),
- $n$: Number of moles of gas,
- $R$: Universal gas constant ($8.31 \, \mathrm{J \, mol^{-1} \, K^{-1}}$),
- $T$: Temperature (measured in kelvin, K).
Key Insights from the Ideal Gas Law:
- Pressure and Volume Relationship:
- Compressing a gas (increasing $P$) reduces its volume ($V$), while reducing the pressure allows the gas to expand, assuming constant temperature and number of moles.
- Temperature and Volume Relationship:
- Heating a gas increases its volume because the particles move faster and exert more outward force.
- Amount of Gas:
- Adding more gas molecules (increasing $n$) increases the pressure or volume, depending on the situation.
- Always convert temperature to kelvin by adding $273.15$ to the Celsius value before using the ideal gas law.
- Kelvin is the absolute temperature scale required for gas law calculations.
A 0.500 m³ tank contains $2.00 \, \mathrm{mol}$ of oxygen gas at a temperature of $300 \, \mathrm{K}$. What is the pressure inside the tank?
Solution
- Write the ideal gas law: $PV = nRT$
- Rearrange for pressure:
$$P = \frac{nRT}{V}$$ - Substitute the known values:
$$P = \frac{(2.00)(8.31)(300)}{0.500}$$ - Calculate:
$$P = 9,972 \, \mathrm{Pa} \, \text{or approximately} \, 10.0 \, \mathrm{kPa}$$
- In this example, we calculated the pressure inside a tank using the ideal gas law.
- Notice how each unit (moles, kelvin, and cubic meters) aligns with the units of $R$.
- This consistency is critical for accurate results.
The Combined Gas Law: Relating Initial and Final States of a Gas
While the ideal gas law is useful for a single set of conditions, many situations involve a gas changing state.
ExampleA balloon might expand as it rises to higher altitudes where the pressure decreases.
The combined gas law relates the initial and final states of a gas:
$$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$$ where:
- $P_1, V_1, T_1$: Initial pressure, volume, and temperature,
