Problem 8 - University Physics with Modern Physics

Question

A welder using a tank of volume 0.0750 \(\mathrm{m}^{3}\) fills it with oxygen \((\text { molar mass } 32.0 \mathrm{g} / \mathrm{mol})\) at a gauge pressure of \(3.00 \times 10^{5} \mathrm{Pa}\) and tenperature of \(37.0^{\circ} \mathrm{C}\) . The tank has a small leak, and in time some of the oxygen leaks out. On a day when the temperature is \(22.0^{\circ} \mathrm{C}\) , the gauge pressure of the oxygen in the tank is \(1.80 \times 10^{5}\) Pa. Find (a) the initial mass of oxygen and (b) the mass of oxygen that has leaked out.

Step-by-Step Solution

Answer

The initial mass of oxygen is calculated, then subtracted by final mass to find the leaked mass.

Step 1: Convert Temperatures to Kelvin

The temperatures given in Celsius need to be converted to Kelvin. The formula for conversion is \( T(K) = T(\degree C) + 273.15 \).For the initial temperature \[ T_1 = 37.0 + 273.15 = 310.15 \, K \].For the final temperature \[ T_2 = 22.0 + 273.15 = 295.15 \, K \].

Step 2: Use Ideal Gas Law to Find Initial and Final Moles

Use the ideal gas law equation \( PV = nRT \) to find the initial and final moles.The pressure values provided are gauge pressures. Since the surrounding atmospheric pressure is approximately \(101,325 \, \text{Pa}\), the absolute pressures are:Initial Pressure: \( P_1 = 3.00 \times 10^5 + 101,325 \) Pa Final Pressure: \( P_2 = 1.80 \times 10^5 + 101,325 \) Pa *A constant needs to be calculated first, \( R = 8.314 \, \text{J/mol} \, K \), and volume remains constant at \( V = 0.0750 \, \text{m}^3 \).First find \( n_1 \): \[ V \cdot P_1 = n_1 \cdot R \cdot T_1 \]\[ n_1 = \frac{V \cdot P_1}{R \cdot T_1} \]Calculate \( n_1 \) using the given values.Then find \( n_2 \): \[ V \cdot P_2 = n_2 \cdot R \cdot T_2 \]\[ n_2 = \frac{V \cdot P_2}{R \cdot T_2} \]Calculate \( n_2 \) using these new values.

Step 3: Calculate Initial Mass of Oxygen

Given that the molar mass of oxygen is \( 32.0 \, \text{g/mol} \), calculate the initial mass:\[ \text{Initial Mass} = n_1 \times 32.0 \, \text{g/mol} \]

Step 4: Calculate Mass of Oxygen that Leaked Out

Find the mass that leaked out by calculating the difference between the initial mass and final mass of oxygen in the tank:Calculate final mass:\[ \text{Final Mass} = n_2 \times 32.0 \, \text{g/mol} \]Finally, the mass leaked is:\[ \text{Mass Leaked} = \text{Initial Mass} - \text{Final Mass} \].

Key Concepts

Pressure and Volume

Understanding the relationship between pressure and volume is crucial, especially in applications involving gases. One of the key principles here is known as Boyle's Law, which states that the pressure of a gas is inversely proportional to its volume, provided the temperature and the amount of gas remain constant. In the Ideal Gas Law, however, we're often dealing with constant volume, like in the case of a fixed tank. This implies changes in pressure due to changes in the number of gas particles or temperature.

In the Ideal Gas Law, represented by the equation \( PV = nRT \), \( P \) is the pressure of the gas, \( V \) is its volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. In scenarios like the original exercise, the volume of the tank is constant (0.0750 \(m^3\)), but the pressure changes due to gas leaking out or temperature shifts. As the pressure changes, it can be calculated using the adjusted volume and temperature, keeping in mind that pressures must include atmospheric pressure for accurate absolute pressure measurements.

By accurately predicting how pressure and volume interact, we can determine the amount of gas remaining in the tank after a leak, indicating the precise relationship highlighted by the Ideal Gas Law.

Temperature Conversion

Temperature plays a pivotal role in calculations involving gases, impacting both pressure and volume according to the Ideal Gas Law. To correctly utilize this law, all temperature measurements need to be in Kelvin, as it is the absolute temperature scale. The conversion from Celsius to Kelvin is straightforward: add 273.15 to the Celsius temperature.

Initially, for an oxygen tank at \(37^{ ext{o}} \text{C}\), the temperature in Kelvin is \(310.15 \text{K}\). After some gas has leaked and the temperature drops to \(22^{ ext{o}} \text{C}\), it converts to \(295.15 \text{K}\). Remember:

• Kelvin eliminates the risk of negative values that can complicate calculations.
• Temperature affects gas particles' energy, influencing both pressure and volume.

Accurate temperature conversion ensures that equations yield reliable results, essential for calculations in physics and chemistry, as wrong temperature values can significantly affect outcomes.

Molar Mass Calculation

Molar mass is a fundamental concept for converting between the number of particles (moles) and mass, specifically in calculations involving gases. The molar mass is the mass of one mole of a substance, measured in grams per mole (\(g/mol\)). For example, oxygen has a molar mass of \(32.0 g/mol\).

In the context of the Ideal Gas Law, once the number of moles of a gas is established using \(n = \frac{PV}{RT}\), you can calculate its mass using:
\[\text{Mass} = n \times \text{Molar Mass}\]
This allows for easy conversion between the moles of gas calculated from the equation and the actual mass, which can be measured or required in practical situations.

• This approach ties together the physical properties and chemical understanding of substances.
• It's crucial for tasks like determining how much gas has leaked, as seen in the exercise, by assessing changes in mass.

Mastering molar mass calculation bridges the gap between theoretical gas laws and tangible, everyday applications in scientific fields.