Problem 13 - General Chemistry: Principles and Modern Applications

Question

A 35.8 L cylinder of \(\mathrm{Ar}(\mathrm{g})\) is connected to an evacuated 1875 L tank. If the temperature is held constant and the final pressure is \(721 \mathrm{mmHg}\), what must have been the original gas pressure in the cylinder, in atmospheres?

Step-by-Step Solution

Answer

The original gas pressure in the cylinder was 50.71 atmospheres.

Step 1: Identify known and unknown variables

The problem provides the following: Initial volume \(V_1\) = 35.8 L, Final volume \(V_2\) = 35.8 + 1875 = 1910.8 L, Final pressure \(P_2\) = 721 mmHg, Temperature \(T\) is constant and Initial pressure \(P_1\) is unknown. The goal is to find \(P_1\).

Step 2: Convert pressure to correct units

The pressure in the problem is given in mmHg, but the standard unit in the ideal gas law is atmospheres. Thus, convert final pressure from mmHg to atmospheres: \(P_2 = 721 mmHg * \frac{1 atm}{760 mmHg} = 0.95 atm.\)

Step 3: Apply Boyle's law

Boyle's law is essentially a restatement of the ideal gas law and states that the pressure and volume of a gas have an inverse relationship when temperature is held constant. Hence, we can write its formula as \(P_1V_1 = P_2V_2\). Substitute the known values into the equation and solve for \(P_1\): \(P_1 = \frac{P_2V_2}{V_1} = \frac{0.95 atm * 1910.8 L}{35.8 L}= 50.71 atm\).

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry that describes how gases behave in different conditions. This law combines several simpler gas laws into one comprehensive equation: \( PV = nRT \), where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume of the gas.
• \( n \) is the number of moles of gas.
• \( R \) is the ideal gas constant.
• \( T \) is the temperature of the gas in Kelvin.

In the problem given, we focus on Boyle's law, a part of the ideal gas law, which assumes that the number of moles and the temperature stay constant. This law is particularly useful when you need to calculate changes in pressure or volume under isothermal (constant temperature) conditions. It's important to remember that pressure and volume are directly connected in this context.

Pressure Conversion

Pressure conversion is crucial in chemistry because different situations or problems may present pressure in varying units. In the exercise, the final pressure was initially given in millimeters of mercury (mmHg), a common measurement in some scientific contexts. However, the standard unit for gas calculations, as used in the ideal gas law, is usually atmospheres (atm).
To convert mmHg to atm, use the conversion factor: \( 1 \text{ atm} = 760 \text{ mmHg} \).
For example, to convert 721 mmHg to atmospheres, you calculate:

• \( 721 \text{ mmHg} \times \frac{1 \text{ atm}}{760 \text{ mmHg}} = 0.95 \text{ atm} \).

This conversion allows us to properly apply Boyle's law or any equations under the ideal gas context. Always ensure to convert pressure to the required unit before substituting values into your equations.

Inverse Relationship

An inverse relationship in mathematics and science means that as one variable increases, the other decreases. When dealing with gases under isothermal conditions, Boyle's law expresses this inverse relationship between pressure and volume. The law is mathematically written as \( P_1V_1 = P_2V_2 \).
Here is how the relationship works:

• If the volume of a gas increases (it has more space to spread out), the pressure decreases (less force exerted by the gas particles on the container walls).
• Conversely, if the volume decreases, the pressure increases as the gas particles collide more frequently with the walls.

This concept highlights the need to maintain a balance or consistency in the total energy and motion of the gas particles when temperature, another critical aspect of the gas environment, remains constant. Understanding this helps you predict how gases will react to changes in their environment.