Understanding Gas Laws and the Ideal Gas Law

The mathematical expression is:

$$P_1 V_1 = P_2 V_2$$

Charles's Law: Volume and Temperature

Charles's Law states that gas volume is directly proportional to its absolute temperature, provided pressure and amount remain constant.

Important Note on Temperature: When working with gas laws, you must always use the absolute temperature scale, known as Kelvin (K). Temperature in Celsius must be converted using the relationship: $T(\text{K}) = T(^\circ\text{C}) + 273.15$.

The concept of absolute zero ($0 \text{ K} = -273.15 ^\circ\text{C}$) represents the theoretical temperature at which the volume of an ideal gas would become zero.

The mathematical expression is:

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

Avogadro's Law: Volume and Moles

Avogadro's Law establishes that the volume of a gas is directly proportional to the amount of gas (the number of moles, $n$) at constant temperature and pressure. Essentially, if you add more particles to a container, the gas will expand to accommodate them.

The mathematical expression is:

$$\frac{V_1}{n_1} = \frac{V_2}{n_2}$$

The Ideal Gas Law

While the individual laws are useful, we can combine them into a single, comprehensive equation known as the Ideal Gas Law. This equation relates all four variables—pressure ($P$), volume ($V$), temperature ($T$), and amount ($n$)—using a single constant called the gas constant ($R$).

$$PV = nRT$$

The value of the gas constant $R$ depends on the units being used:

• Using Liters and Atmospheres: $R = 0.08206 \text{ L}\cdot\text{atm}/(\text{mol}\cdot\text{K})$
• Using SI Units: $R = 8.314 \text{ J}/(\text{mol}\cdot\text{K})$

Exam Tip: Always check your units before plugging numbers into the Ideal Gas Law. If your pressure is in kPa, you must use the corresponding $R$ value or convert your pressure to atm first!

Worked Example: Finding Moles

Question: An automobile tire at $23^\circ\text{C}$ with an internal volume of $25.0 \text{ L}$ is filled with air to a total pressure of $3.18 \text{ atm}$. Determine the number of moles of air in the tire.

Solution:

1. Convert Temperature to Kelvin: $23 + 273 = 296 \text{ K}$
2. Rearrange the Ideal Gas Law for $n$:

$$n = \frac{PV}{RT}$$

3. Substitute and Solve:

$$n = \frac{(3.18 \text{ atm})(25.0 \text{ L})}{(0.08206 \frac{\text{L}\cdot\text{atm}}{\text{mol}\cdot\text{K}})(296 \text{ K})} = 3.27 \text{ mol}$$

Gas Density and Molar Mass

We can extend the Ideal Gas Law to understand the physical properties of a gas, such as its density ($\rho$). Density is defined as mass ($m$) divided by volume ($V$). By substituting the relationship between moles and mass ($n = m/M$, where $M$ is the molar mass) into the Ideal Gas Law, we derive:

$$PV = \left(\frac{m}{M}\right)RT$$

Rearranging for density ($\rho = m/V$):

$$\rho_{\text{gas}} = \frac{PM}{RT}$$

From this formula, we can observe three critical relationships regarding gas density:

1. Pressure: Density is directly proportional to pressure (higher pressure = denser gas).
2. Temperature: Density is inversely proportional to temperature (higher temperature = less dense gas).
3. Molar Mass: Density is directly proportional to molar mass (heavier molecules = denser gas).