The Ideal Gas Law: Videos & Practice Problems

Ideal gases are simplified models that assume low density, no intermolecular forces, zero physical size, and elastic collisions. The ideal gas law, expressed as $P^V=nRT$ or $P^V=NkT$, relates pressure, volume, temperature, and moles. At standard temperature and pressure (STP), one mole occupies 22.4 liters. Boyle's law illustrates the inverse relationship between pressure and volume when temperature is constant, highlighting fundamental gas behavior.

Ideal Gases and the Ideal Gas Law

In the study of thermodynamics, the concept of ideal gases plays a crucial role. An ideal gas is a theoretical gas that perfectly follows the ideal gas law, which simplifies the behavior of gases under various conditions. The ideal gas law is expressed as PV = nRT, where P represents pressure, V is volume, n is the number of moles, R is the universal gas constant (8.314 J/(mol·K)), and T is the absolute temperature in Kelvin.

To understand ideal gases, it is essential to recognize the conditions they satisfy. First, ideal gases have low density, meaning their particles are widely spaced, resulting in low pressure and high temperature. Second, there are no intermolecular forces acting between the gas particles, which allows them to behave independently. Third, the particles are considered to have no physical size, treated as point particles. Lastly, they move in straight lines and collide elastically, conserving energy during collisions.

In practical terms, many real gases behave similarly to ideal gases under standard conditions, such as those found in the Earth's atmosphere. The ideal gas law can be applied to any ideal gas, and there are two forms of the equation: PV = nRT for moles and PV = Nk_bT for the number of particles, where N is the number of particles and k_b is Boltzmann's constant (1.38 x 10^-23 J/K).

When using the ideal gas law, it is crucial to work with absolute temperatures, which must be expressed in Kelvin. A common reference point in gas calculations is Standard Temperature and Pressure (STP), defined as 0 degrees Celsius (273 K) and 1 atmosphere (1.01 x 10⁵ Pa).

For example, to calculate the volume of one mole of an ideal gas at STP, we can rearrange the ideal gas law to find V: V = nRT/P. Substituting the known values (n = 1 mol, R = 8.314 J/(mol·K), T = 273 K, and P = 1.01 x 10⁵ Pa), we find that the volume is approximately 0.0224 m³ or 22.4 liters. This volume is significant as it represents the molar volume of any ideal gas at STP, illustrating that one mole of an ideal gas occupies the same volume under these conditions, regardless of the gas type.

3 moles of an ideal gas fill a cubical box with a side length of 30cm. If the temperature of the gas is 20°C, what is the pressure inside the container?

• A. 2.7 × 10⁵ Pa
• B. 0.27 Pa
• C. 1.85 × 10⁴ Pa
• D. 8.12 × 10³ Pa

Answer: (A)

Hydrogen gas behaves very much like an ideal gas. If you have a sample of Hydrogen gas with a volume of 1000 cm³ at 30°C with a pressure of 1 × 10⁵ Pa, calculate how many hydrogen atoms (particles) there are in the sample.

• A. 2.42 × 10²²
• B. 2.39 × 10²²
• C. 2.39 × 10²⁸

Answer: (B)

Solving Ideal Gas Problems With Changing States

The ideal gas law, represented by the equation PV = nRT, describes the relationship between pressure (P), volume (V), number of moles (n), the universal gas constant (R), and temperature (T) of an ideal gas. When analyzing changes in a gas's state, such as during compression, it is essential to compare the initial and final states of the gas. This involves using the ideal gas law to establish a relationship between the variables at these two states.

To compare the initial and final states, we can express the ideal gas law for both conditions: P_initialV_initial = n_initialRT_initial and P_finalV_final = n_finalRT_final. By setting these two equations equal to each other, the constant R cancels out, leading to the simplified equation: \(\frac{P_{initial} V_{initial}}{n_{initial} T_{initial}} = \frac{P_{final} V_{final}}{n_{final} T_{final}}\). This equation is crucial for solving problems involving changes in the state of an ideal gas.

In many scenarios, two of the four variables (P, V, n, T) remain constant, allowing for further simplification. For instance, if the number of moles remains constant (n = 2) and the temperature is constant (T_initial = T_final), these variables can be canceled out from the equation. This leaves a relationship between the initial and final pressures and volumes, which can be expressed as P_initialV_initial = P_finalV_final.

For example, if the initial pressure is 1 × 10⁵ Pa and the initial volume is 0.05 m³, while the final volume is 0.01 m³, we can rearrange the equation to find the final pressure: P_final = \(\frac{P_{initial} V_{initial}}{V_{final}}\). Plugging in the values gives us a final pressure of 5 × 10⁵ Pa.

This relationship illustrates Boyle's Law, which states that for a given amount of gas at constant temperature, pressure is inversely proportional to volume. In other words, as volume decreases, pressure increases, and vice versa. This principle is a specific case of the ideal gas law.

Additionally, there are other gas laws derived from the ideal gas law. Charles' Law states that at constant pressure, volume is directly proportional to temperature, while Gay-Lussac's Law states that at constant volume, pressure is directly proportional to temperature. Understanding these relationships is essential for solving various gas-related problems in physics and chemistry.

A balloon contains 3900cm³ of a gas at a pressure of 101 kPa and a temperature of –9°C. If the balloon is warmed such that the temperature rises to 28°C, what volume will the gas occupy? Assume the pressure remains constant.

• A. 0.012 m³
• B. 0.0041 m³
• C. 0.39 m³
• D. 0.0044 m³

Answer: (D)

Doubling Pressure & Temperature

In this scenario, we are analyzing the behavior of an ideal gas contained in a sealed container where both the pressure and volume are doubled. The goal is to determine the final temperature of the gas after these changes occur.

To begin, we utilize the ideal gas law, which is expressed as:

PV = nRT

In this equation, P represents pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. For a sealed container, the number of moles remains constant, allowing us to set up the relationship:

P_{initial} V_{initial} / T_{initial} = P_{final} V_{final} / T_{final}

Given that both pressure and volume double, we can express the final conditions as:

P_{final} = 2P_{initial}

V_{final} = 2V_{initial}

Substituting these values into the equation, we simplify to find:

T_{final} = (P_{final} V_{final} / (P_{initial} V_{initial})) T_{initial}

By substituting the expressions for P and V, we find:

T_{final} = (2P_{initial} * 2V_{initial}) / (P_{initial} V_{initial}) * T_{initial} = 4T_{initial}

This indicates that the final temperature is four times the initial temperature. If the initial temperature is 40 degrees Celsius, converting this to Kelvin gives:

T_{initial} = 40 + 273 = 313 K

Thus, the final temperature becomes:

T_{final} = 4 * 313 = 1252 K

Next, we need to determine the number of moles of gas in the container. We can use the ideal gas law again, either with initial or final conditions, since the number of moles remains constant. Using the initial conditions:

n = P_{initial} V_{initial} / (R T_{initial})

Here, we need to ensure that the units are consistent. The initial pressure is given as 0.15 atmospheres, which we convert to Pascals:

P_{initial} = 0.15 * 1.01 \times 10^5 = 15150 Pa

The initial volume is given as 2.8 liters, which we convert to cubic meters:

V_{initial} = 2.8 * 0.001 = 0.0028 m^3

Now substituting these values into the equation for n:

n = (15150 * 0.0028) / (8.314 * 313)

Calculating this gives:

n ≈ 0.016 moles

In summary, after the changes in pressure and volume, the final temperature of the gas is 1252 K, and the number of moles of gas in the container is approximately 0.016 moles.

Here’s what students ask on this topic:

What is the ideal gas law and how is it used?

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure (P), volume (V), temperature (T), and amount of substance (n) of an ideal gas. It is expressed as:

$PV=nRT$

where R is the universal gas constant (8.314 J/(mol·K)). This law is used to calculate any one of the four variables if the other three are known. It is particularly useful in predicting the behavior of gases under different conditions, such as changes in pressure, volume, or temperature.

What are the conditions for a gas to be considered ideal?

A gas is considered ideal if it meets the following conditions: (1) Low density, meaning particles are spread out and not tightly packed. (2) No intermolecular forces between particles. (3) Particles have zero physical size, treated as point particles. (4) Particles move in straight lines and collide elastically, conserving energy. These assumptions simplify the behavior of gases, making it easier to apply the ideal gas law.

How do you convert between the two forms of the ideal gas law?

The two forms of the ideal gas law are:

$PV=nRT$

and

$PV=NkBT$

where N is the number of particles and k_B is Boltzmann's constant (1.38 × 10^-23 J/K). To convert between them, use the relationship:

$n=\frac{N}{N}$

where N_A is Avogadro's number (6.02 × 10²³ particles/mol).

What is the significance of standard temperature and pressure (STP) in the ideal gas law?

Standard Temperature and Pressure (STP) is a reference point used in gas calculations. It is defined as a temperature of 0°C (273 K) and a pressure of 1 atmosphere (1.01 × 10⁵ Pascals). At STP, one mole of an ideal gas occupies 22.4 liters. This standardization allows for easy comparison and calculation of gas properties under consistent conditions.

How do Boyle's, Charles', and Gay-Lussac's laws relate to the ideal gas law?

Boyle's, Charles', and Gay-Lussac's laws are special cases of the ideal gas law:

1. Boyle's Law: At constant temperature, pressure is inversely proportional to volume (P ∝ 1/V).

2. Charles' Law: At constant pressure, volume is directly proportional to temperature (V ∝ T).

3. Gay-Lussac's Law: At constant volume, pressure is directly proportional to temperature (P ∝ T).

These laws describe the behavior of gases under specific conditions and can be derived from the ideal gas law by holding certain variables constant.