Question

If \(2.51 \mathrm{g}\) of \(\mathrm{H}_{2}\) is placed in an evacuated \(255-\mathrm{mL}\) container at \(100 .^{\circ} \mathrm{C},\) what will be the pressure inside the container?

Short answer

To find the pressure inside the container, first calculate the number of moles of H₂: \(n = \frac{2.51 \mathrm{g}}{2 \mathrm{g/mol}} = 1.255 \mathrm{mol}\). Convert the temperature to Kelvin: \(100^\circ\mathrm{C} + 273.15 = 373.15\mathrm{K}\) and the volume to liters: \(255\mathrm{mL} \div 1000 = 0.255\mathrm{L}\). Now, use the ideal gas law formula: \(P = \frac{nRT}{V} = \frac{1.255 \mathrm{mol} \cdot 0.0821 \frac{\mathrm{L\cdot atm}}{\mathrm{mol\cdot K}} \cdot 373.15\mathrm{K}}{0.255\mathrm{L}} = \text{approximately} 15.89\mathrm{atm}\). The pressure inside the container is approximately \(15.89\mathrm{atm}\).

Step-by-step solution

Find the number of moles of Hydrogen gas

For this step, we need the molar mass of Hydrogen gas (H₂). The molar mass of Hydrogen (H) is approximately 1 g/mol. As there are two Hydrogen atoms in H₂, the molar mass of H₂ will be 2 g/mol. The mass of H₂ given is 2.51 g. We can now find the number of moles (n) by dividing the mass by the molar mass: n = (mass of H₂) / (molar mass of H₂) n = 2.51 g / 2 g/mol

Convert the temperature to Kelvin & convert mL to L

The ideal gas law requires the temperature to be in Kelvin (K). So, we need to convert the given temperature in Celsius (100°C) to Kelvin, using the formula: Temperature (K) = Temperature (°C) + 273.15 We also need the volume to be in Liters (L), so we will convert the given volume from milliliters (mL) to liters: Volume (L) = Volume (mL) / 1000

Calculate the pressure using the ideal gas law

Now that we have the number of moles, the volume in liters, and the temperature in Kelvin, we can solve for the pressure inside the container using the ideal gas law, which is PV = nRT. In this formula, R is the ideal gas constant, which is approximately 0.0821 L·atm/mol·K. Rearranging the formula to find the pressure (P), we get: P = nRT / V Now plug in the values and calculate the pressure: P = (n) * (0.0821 L·atm/mol·K) * (Temperature in K) / (Volume in L)

Write the final answer

After solving the equation in step 3, we will get the pressure inside the container in atmospheres (atm).

Key concepts

Molar Mass

Understanding the concept of molar mass is crucial in chemistry, especially when dealing with the ideal gas law. Molar mass is defined as the mass of one mole of a substance, measured in grams per mole (\text{g/mol}). It corresponds to the mass of 6.02 \( \times \) 10^{23} particles of that substance, which is Avogadro's number.

For example, the molar mass of hydrogen (\text{H}_2) can be calculated by adding the masses of two hydrogen atoms. Since the atomic mass of one hydrogen atom is approximately 1 gram per mole, the molar mass of hydrogen gas (\text{H}_2) becomes 2 g/mol. When solving problems, the accurate molar mass allows us to convert mass into moles, which is a pivotal step in using the ideal gas law.

Moles of Gas

In chemistry, the mole is a fundamental unit that measures the amount of substance. Moles of gas refer to the number of moles present in a given volume of gas. It's vital to calculate this quantity as it ties the mass of a gas to its molar mass and further provides a link to its behavior under certain conditions as described by gas laws.

To find the moles of gas, we divide the mass of the gas by its molar mass, applying this formula: \( n = \frac{\text{mass of gas}}{\text{molar mass}} \). For example, if we have 2.51 g of \text{H}_2, we divide it by the molar mass of \text{H}_2 (2 g/mol) to find the moles of gas.

Gas Laws

The behavior of gases and their physical properties are explained by various gas laws. One fundamental relation is the ideal gas law, represented by the equation PV = nRT. This law correlates the pressure (P), volume (V), number of moles (n), and temperature (T) of an ideal gas to the ideal gas constant (R).

The ideal gas law assumes no forces between gas molecules and that the molecules occupy no volume, which is a close approximation for real gases under many conditions. Understanding this law is essential for solving problems related to gas pressures, volumes, and temperatures.

Temperature Conversion

To use the ideal gas law effectively, it's necessary to convert the temperature into Kelvin (K). This is because the Kelvin scale is an absolute temperature scale used in scientific equations to ensure consistency.

The conversion from Celsius to Kelvin is straightforward: add 273.15 to the Celsius temperature. So, a temperature of 100°C becomes 373.15 K. When solving gas law problems, it's imperative to remember to make this conversion, as using Celsius or Fahrenheit will not yield correct results.

Pressure Calculation

Pressure is one of the critical variables in the ideal gas law and can be defined as the force applied per unit area. In gas law problems, calculating gas pressure involves rearranging the ideal gas equation to solve for pressure (P = \( \frac{nRT}{V} \)). For example, once you have the number of moles (n), the absolute temperature (T), and the volume (V), you can find the pressure exerted by the gas.

It is important to keep units consistent, hence why standard units such as liters for volume and atmospheres for pressure are used. This ensures that the gas constant (\text{R}), with its specific units, can be correctly applied to obtain the pressure in atmospheres (atm).