Question

Explain how Boyle’s law, Charles’s law, and Avogadro’s law are special cases of the ideal gas law

Short answer

Boyle's Law, Charles's Law, and Avogadro's Law are special cases of the ideal gas law, with the equation \( PV = nRT \). Boyle's Law is a special case when temperature and the number of moles are constant, so \( P_1V_1 = P_2V_2 \). Charles's Law is a special case when pressure and the number of moles are constant, resulting in \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \). Avogadro's Law is a special case when temperature and pressure are constant, giving \( \frac{V_1}{n_1} = \frac{V_2}{n_2} \).

Step-by-step solution

Write down the ideal gas law equation

The ideal gas law equation is given by: \( PV = nRT \) where P is the pressure, V is the volume, n is the number of moles of the gas, R is the universal gas constant, and T is the temperature in Kelvin.

Analyze Boyle's Law

Boyle's Law states that for a fixed amount of an ideal gas at constant temperature (i.e., when n and T are held constant), the pressure and volume of the gas are inversely proportional. Mathematically, this can be expressed as: \( P_1V_1 = P_2V_2 \)

Show Boyle's Law as a special case of the ideal gas law

Since n and T are constant in Boyle's Law, we can rewrite the ideal gas law equation as: \( PV = constant \) Now, let's substitute the initial pressure and volume as \( P_1 \) and \( V_1 \) and final pressure and volume as \( P_2 \) and \( V_2 \) in the ideal gas law equation: \( P_1V_1 = nRT \) and \( P_2V_2 = nRT \) Since both equations are equal to \( nRT \), we can equate the left-hand sides of the two equations: \( P_1V_1 = P_2V_2 \) Thus, Boyle's Law is a special case of the ideal gas law when the temperature and the number of moles are held constant.

Analyze Charles's Law

Charles's Law states that for a fixed amount of an ideal gas at constant pressure (i.e., when n and P are held constant), the volume and temperature of the gas are directly proportional. Mathematically, this can be expressed as: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \)

Show Charles's Law as a special case of the ideal gas law

Since n and P are constant in Charles's Law, we can rewrite the ideal gas law equation as: \( V = \frac{nRT}{P} \) Now, let's substitute the initial volume and temperature as \( V_1 \) and \( T_1 \) and the final volume and temperature as \( V_2 \) and \( T_2 \) in the ideal gas law equation: \( V_1 = \frac{nRT_1}{P} \) and \( V_2 = \frac{nRT_2}{P} \) Dividing the first equation by the second equation gives: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \) Thus, Charles's Law is a special case of the ideal gas law when the pressure and the number of moles are held constant.

Analyze Avogadro's Law

Avogadro's Law states that for a gas at constant temperature and pressure (i.e., when T and P are held constant), the volume of the gas is directly proportional to the number of moles. Mathematically, this can be expressed as: \( \frac{V_1}{n_1} = \frac{V_2}{n_2} \)

Show Avogadro's Law as a special case of the ideal gas law

Since T and P are constant in Avogadro's Law, we can rewrite the ideal gas law equation as: \( V = \frac{nRT}{P} \) Now, let's substitute the initial volume and moles as \( V_1 \) and \( n_1 \) and the final volume and moles as \( V_2 \) and \( n_2 \) in the ideal gas law equation: \( V_1 = \frac{n_1RT}{P} \) and \( V_2 = \frac{n_2RT}{P} \) Dividing the first equation by the second equation gives: \( \frac{V_1}{n_1} = \frac{V_2}{n_2} \) Thus, Avogadro's Law is a special case of the ideal gas law when the temperature and pressure are held constant.

Key concepts

Boyle's Law

Boyle's Law beautifully explains how the pressure of a gas relates to its volume when the temperature and the number of gas moles stay the same. Imagine a balloon. When you squeeze it, the volume reduces, and the pressure inside increases. This is Boyle's Law in action, encapsulating the concept of inverse proportionality between pressure and volume. If gas is compressed into a smaller volume, its pressure increases proportionally. In the mathematical world, this is expressed as:

• Initial state: \( P_1V_1 \)
• Final state: \( P_2V_2 \)

This shows that \( P_1V_1 = P_2V_2 \) when temperature and the number of moles are constant, defining Boyle's Law as a special instance under the overarching ideal gas law.

Charles's Law

Charles's Law uncovers the relationship between the volume and temperature of a gas when pressure and amount of gas remain fixed. Picture a hot air balloon. As the air inside heats up, it expands. This demonstrates Charles's Law, where the volume of gas expands with increased temperature, provided the amount and pressure stay the same. Mathematically, this translates to:

• Initial state: \( \frac{V_1}{T_1} \)
• Final state: \( \frac{V_2}{T_2} \)

Therefore, it is concluded that \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \). This relationship highlights Charles's Law fitting snugly as a specific case of the ideal gas law when pressure and moles are constant.

Avogadro's Law

Avogadro's Law dives into how the amount of gas moles influences its volume at steady temperature and pressure. Consider inflating a tire. The more air you pump in, the larger the tire becomes. This is Avogadro's Law coming into play, where the volume increases with the number of gas moles, assuming constant temperature and pressure. Expressed mathematically, we have:

• Initial state: \( \frac{V_1}{n_1} \)
• Final state: \( \frac{V_2}{n_2} \)

It then stands that \( \frac{V_1}{n_1} = \frac{V_2}{n_2} \). Thus, Avogadro's Law reveals itself as a salient facet of the ideal gas law when temperature and pressure are held stable.