Question

Suppose two \(200.0-\mathrm{L}\) tanks are to be filled separately with the gases helium and hydrogen. What mass of each gas is needed to produce a pressure of 135 atm in its respective tank at \(24^{\circ} \mathrm{C}\) ?

Short answer

To fill the two $200.0-L$ tanks with helium and hydrogen, respectively, to a pressure of 135 atm at $24^{\circ} C$, we first find the number of moles using the ideal gas law, \(PV = nRT\). Upon calculating, we find that the number of moles is the same for both gases. Next, we determine the mass of each gas using their molar masses: For helium: \(m_{He} = n \cdot 4.00 \frac{g}{mol}\) For hydrogen: \(m_{H_2} = n \cdot 2.02 \frac{g}{mol}\) These expressions provide the required masses of helium and hydrogen needed to fill the tanks at the given pressure and temperature.

Step-by-step solution

Convert temperature and pressure to proper units

First, we need to convert the given temperature from Celsius to Kelvin: \[T(K) = 24^{\circ} C + 273.15 = 297.15 K\] We also have the pressure given in atm, which is an acceptable unit for this problem.

Calculate the number of moles for each gas

Now we'll use the ideal gas law, \(PV = nRT\), to calculate the number of moles for helium and hydrogen. We are given the pressure, volume, and temperature, and know the gas constant \(R = 0.0821\frac{L\cdot atm}{mol\cdot K}\). For helium: \[n_{He} = \frac{PV}{RT} = \frac{135 atm \cdot 200.0 L}{0.0821\frac{L\cdot atm}{mol\cdot K} \cdot 297.15 K}\] For hydrogen: \[n_{H_2} = \frac{PV}{RT} = \frac{135 atm \cdot 200.0 L}{0.0821\frac{L\cdot atm}{mol\cdot K} \cdot 297.15 K}\] As we can see, the number of moles for both gases is the same, so we just need to calculate it once. \[n = \frac{135 \cdot 200.0}{0.0821 \cdot 297.15}\]

Calculate the mass of each gas using their molar masses

Use the molar masses of helium and hydrogen to find the mass of each gas: Molar mass of helium: \(4.00 \frac{g}{mol}\) Molar mass of hydrogen: \(2.02 \frac{g}{mol}\) Mass of helium: \[m_{He} = n_{He} \cdot M_{He} = n \cdot 4.00 \frac{g}{mol}\] Mass of hydrogen: \[m_{H_2} = n_{H_2} \cdot M_{H_2} = n \cdot 2.02 \frac{g}{mol}\] Now, let's calculate the mass of each gas: For helium: \[m_{He} = n \cdot 4.00 \frac{g}{mol}\] For hydrogen: \[m_{H_2} = n \cdot 2.02 \frac{g}{mol}\]

Key concepts

Converting Celsius to Kelvin

When working with the ideal gas law, temperature must be in Kelvin. This is because Kelvin is the absolute temperature scale and the ideal gas law is derived on the assumption that temperature is measured from absolute zero, where all thermal motion ceases.

To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. This is because the size of one degree in Kelvin is the same as one degree in Celsius, but the scales have different starting points. The Kelvin scale starts at absolute zero, which is -273.15°C. Therefore, the formula for the conversion is:

\[ T(K) = T(°C) + 273.15 \]

For example, in our exercise, the given temperature of 24°C would be converted as follows:

\[ 24°C + 273.15 = 297.15 K \]

This step is crucial because if you mistakenly use Celsius in the ideal gas law calculations, you would get incorrect results. It's an easy but critical adjustment to make before proceeding with further calculations.

Calculating Moles Using Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as \[ PV = nRT \]

Here,

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

To find the number of moles (\( n \)), rearrange the formula to solve for \( n \):

\[ n = \frac{PV}{RT} \]

With the pressure (\( P \)) and volume (\( V \)) given, alongside the temperature (\( T \)) already converted to Kelvin, and the ideal gas constant (\( R \)) known, you can calculate the number of moles. For both helium and hydrogen in our exercise, this step is the same since the conditions are identical for both gases.

Molar Mass of Gases

The molar mass of a gas is the mass of one mole of that gas. It is a physical property unique to each substance and is especially important when you need to convert between the amount of substance (in moles) and its corresponding mass in grams.

The molar mass is typically given in units of grams per mole \(\left(\frac{g}{mol}\right)\). To find the mass of a gas using the number of moles calculated with the ideal gas law, you multiply the number of moles by the molar mass of the gas:

\[ m = n \cdot M \]

In this formula:

• \( m \) represents the mass of the gas,
• \( n \) is the number of moles, and
• \( M \) is the molar mass of the gas.

For our given problem involving helium and hydrogen, knowing their molar masses (4.00 g/mol for helium and 2.02 g/mol for hydrogen) allows us to find the mass of each gas needed to fill the tanks.

Pressure-Volume-Temperature Relationship

The pressure-volume-temperature relationship is a central part of the ideal gas law. This relationship shows that for a fixed amount of gas at a constant temperature, the pressure is inversely proportional to the volume, and for a given pressure and volume, the temperature must be proportional to the amount of the gas present.

The ideal gas law encapsulates this relationship: as the volume (\( V \)) of the gas increases, the pressure (\( P \)) it exerts decreases, and vice versa, as long as the temperature (\( T \)) and number of moles (\( n \)) stay constant. Similarly, if the temperature increases while keeping the pressure constant, the volume must increase to accommodate the higher kinetic energy of the gas molecules.

These relationships are crucial in practical applications, like filling tanks with gases under certain conditions, as illustrated in our exercise. Understanding these principles helps ensure safe and efficient practices in various scientific and industrial processes.