Problem 78 - Chemistry The Science in Context

Question

Hydrogen holds promise as an "environment friendly" fuel. How many grams of \(\mathrm{H}_{2}\) gas are present in a \(50.0 \mathrm{L}\) fuel tank at a pressure of \(2850 \mathrm{lb} / \mathrm{in}^{2}(\mathrm{psi})\) at \(20^{\circ} \mathrm{C} ?\) Assume that \(1 \mathrm{atm}=14.7 \mathrm{psi}\).

Step-by-Step Solution

Answer

Answer: The mass of hydrogen gas in the fuel tank is approximately 86.8 grams.

Step 1: Convert pressure to atmospheres

To obtain the pressure in atmospheres, divide the given pressure of 2850 psi by the conversion factor 14.7 psi/atm: Pressure in atm = \(\frac{2850 \text{ psi}}{14.7 \text{ psi/atm}}\) = 193.88 atm

Step 2: Convert the temperature to Kelvin

As the temperature is given in Celsius, we need to convert it to Kelvin by adding 273.15 K to the given temperature: Temperature in K = \(20^{\circ} \mathrm C + 273.15 \mathrm K = 293.15 \mathrm K\)

Step 3: Use the Ideal Gas Law to find the number of moles of hydrogen gas

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of gas as follows: PV=nRT Where P is the pressure in atm, V is the volume in liters, n is the number of moles, R is the gas constant (0.0821 L⋅atm/mol⋅K), and T is the temperature in Kelvin. Rearranging the equation to solve for the number of moles (n) and plugging in the values from steps 1 and 2: n = \(\frac{PV}{RT} = \frac{(193.88 \text{ atm})(50.0 \, \text{L})}{(0.0821 \frac{\text{L}\cdot\text{atm}}{\text{mol}\cdot\text{K}})(293.15 \, \text{K})} = 42.973 \, \text{mol}\)

Step 4: Calculate the mass of hydrogen gas

To find the mass of hydrogen gas, multiply the number of moles calculated in step 3 by the molar mass of hydrogen gas (approximately 2.02 g/mol): Mass = number of moles × molar mass of hydrogen = \(42.973 \, \text{mol} \times 2.02 \frac{\text{g}}{\text{mol}} = 86.805 \, \text{g}\) Therefore, there are approximately 86.8 grams of hydrogen gas present in the 50.0 L fuel tank.

Key Concepts

Pressure Conversion

To solve problems involving gases, it's crucial to convert pressure units correctly. In this exercise, pressure is given in pounds per square inch (psi), but for calculations involving the Ideal Gas Law, it's beneficial to use atmospheres (atm) as a standard unit. This is due to the gas constant value that typically uses atm.

• Pressure in psi: 2850 psi
• Conversion factor: 1 atm = 14.7 psi

To convert the pressure to atm, divide the given pressure by the conversion factor:\[\text{Pressure in atm} = \frac{2850 \text{ psi}}{14.7 \text{ psi/atm}} = 193.88 \text{ atm}\]This conversion is important for accurate use of the gas constant in later calculations.

Temperature Conversion

Temperature must be in Kelvin when using the Ideal Gas Law, because this law requires an absolute temperature scale. In this question, the temperature is initially in degrees Celsius.

• Temperature in Celsius: \(20^{\circ} \text{C}\)
• Conversion: Add 273.15 to the Celsius temperature to convert it to Kelvin.

Thus:\[\text{Temperature in K} = 20^{\circ} \text{C} + 273.15 = 293.15 \text{ K}\]Converting to Kelvin ensures that the proportionality to absolute temperature is maintained, which is necessary for calculations using gas laws.

Mole Calculation

The Ideal Gas Law, \(PV = nRT\), allows us to determine the number of moles of a gas when we know the pressure, temperature, and volume.

• \(P\) (pressure) = 193.88 atm
• \(V\) (volume) = 50.0 L
• \(R\) (gas constant) = 0.0821 L⋅atm/mol⋅K
• \(T\) (temperature) = 293.15 K

To find the number of moles \(n\), rearrange the Ideal Gas Law:\[n = \frac{PV}{RT} = \frac{(193.88 \text{ atm})(50.0 \text{ L})}{(0.0821 \frac{\text{L}\cdot\text{atm}}{\text{mol}\cdot\text{K}})(293.15 \text{ K})} = 42.973 \text{ mol}\]This calculation gives us the amount of substance in moles, which is a crucial step before determining the mass.

Molar Mass Calculation

Once the number of moles is known, converting this to mass requires the molar mass of the substance. For hydrogen gas (\(\text{H}_2\)), the molar mass can be deduced from the atomic mass of hydrogen.

• Molar mass of hydrogen (\(\text{H}_2\)): Approximately 2.02 g/mol

Using this molar mass, the mass of hydrogen gas is:\[\text{Mass} = \text{number of moles} \times \text{molar mass} = 42.973 \text{ mol} \times 2.02 \frac{\text{g}}{\text{mol}} = 86.805 \text{ g}\]By using the molar mass, we convert the moles derived from the Ideal Gas Law into an understandable mass in grams, showing us exactly how much hydrogen is in the tank.