Problem 12 - Physics

Question

Oxygen for hospital patients is kept in special tanks, where the oxygen has a pressure of 65.0 atmospheres and a temperature of \(288 \mathrm{~K}\). The tanks are stored in a separate room, and the oxygen is pumped to the patient's room, where it is administered at a pressure of 1.00 atmosphere and a temperature of \(297 \mathrm{~K}\). What volume does \(1.00 \mathrm{~m}^{3}\) of oxygen in the tanks occupy at the conditions in the patient's room?

Step-by-Step Solution

Answer

The oxygen occupies approximately 67.03 cubic meters in the patient's room.

Step 1: Understanding the Problem

We need to find the volume of oxygen at different conditions using the ideal gas law. Initially, the gas is in a tank with a certain pressure and temperature, and we will find its volume under different conditions in the patient's room.

Step 2: Apply the Ideal Gas Law

We will use the combined gas law in the form \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \), where \(P_1, V_1, T_1\) are the initial pressure, volume, and temperature, and \(P_2, V_2, T_2\) are the final conditions. Given: \(P_1 = 65.0\, \text{atm}\), \(V_1 = 1.00\, \text{m}^3\), \(T_1 = 288\, \text{K}\), \(P_2 = 1.00\, \text{atm}\), \(T_2 = 297\, \text{K}\).

Step 3: Solve for Final Volume

Rearrange the combined gas law to solve for \(V_2\): \[ V_2 = V_1 \times \frac{P_1}{P_2} \times \frac{T_2}{T_1} \]Substitute the given values:\[ V_2 = 1.00 \, \text{m}^3 \times \frac{65.0}{1.00} \times \frac{297}{288} \]

Step 4: Calculate Final Volume

Calculate the expression from Step 3:\[ V_2 = 1.00 \, \text{m}^3 \times 65.0 \times 1.03125 = 67.03125 \, \text{m}^3 \]Thus, the volume of the oxygen at the conditions of the patient's room is approximately 67.03 cubic meters.

Key Concepts

Combined Gas Law

The combined gas law is an extremely useful formula which helps us understand how gases behave under changing conditions. It combines three fundamental gas laws: Boyle's Law, Charles' Law, and Gay-Lussac's Law. In essence, it allows us to predict how a gas will behave when its pressure, volume, and temperature change all at once.
To make use of the combined gas law, we apply the formula: \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]

• \(P_1\) and \(P_2\) stand for the initial and final pressure.
• \(V_1\) and \(V_2\) are the initial and final volume.
• \(T_1\) and \(T_2\) represent the initial and final temperatures.

By using this equation, you can solve for any one of the variables if you know the others. The law assumes the amount of gas remains constant. Its balanced use ensures that while one aspect such as pressure might increase, others like volume might decrease to maintain consistency. By understanding this relationship, we gain a better grasp of gas behavior in different situations, like the exercise involving hospital oxygen tanks.

Pressure and Volume Relationship

When examining gases, the pressure and volume relationship is vital. This relationship is described by Boyle's Law, which states that, provided the temperature is constant, the pressure of a gas is inversely proportional to its volume.
In simpler terms:

• If the pressure increases, then the volume decreases.
• If the pressure decreases, then the volume increases.

The formula of Boyle's Law is:\[ P_1 V_1 = P_2 V_2 \]This is very relevant in scenarios such as the oxygen tanks in hospitals. The tanks store oxygen at a high pressure and relatively low volume. When it's time to deliver this oxygen to a patient at a lower pressure, the volume expands to ensure the required amount reaches the recipient effectively.
This relationship is foundational when using the combined gas law, where we notice that, by changing the pressure from 65 to 1 atmospheres, the volume was found to increase significantly from 1.00 cubic meter to over 67 cubic meters. This demonstrates the inverse connection between pressure and volume.

Temperature and Volume Relationship

The temperature and volume relationship of gases is explained by Charles' Law. It states that the volume of a gas is directly proportional to its temperature, assuming the pressure remains constant. This means:

• If the temperature of a gas increases, its volume increases.
• If the temperature decreases, its volume decreases.

Expressed mathematically, Charles' Law appears as:\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]This relationship highlights how gases expand or contract with temperature shifts. It's crucial in the context of the combined gas law as well. In the provided exercise, as the oxygen's temperature rises slightly from 288 K to 297 K during the transfer from the tank to the patient's room, we notice a corresponding increase in volume.
With the temperature change factored in together with a pressure change, it explains why oxygen that occupied just one cubic meter in high-pressure conditions expands significantly to fill 67 cubic meters once conditions adapted.