Question

At what temperature does \(4.00 \mathrm{g}\) of helium gas have a pressure of 1.00 atm in a 22.4 -L vessel?

Short answer

The temperature at which 4.00 g of helium gas has a pressure of 1.00 atm in a 22.4 L vessel is approximately 273 K.

Step-by-step solution

Calculate the number of moles of helium

To find the number of moles of helium, we need to divide the mass of helium (4.00g) by its molar mass (4.00 g/mol). n = mass / molar_mass n = 4.00 g / (4.00 g/mol) = 1.00 mol

Use the ideal gas law to find the temperature

The ideal gas law equation is: PV = nRT Where: P = pressure = 1.00 atm V = volume = 22.4 L n = moles of gas = 1.00 mol R = ideal gas constant = 0.0821 L*atm/mol*K (for calculations in atm and L) We need to find T (temperature). Rearrange the ideal gas law to solve for T: T = PV/(nR) Plug in the known values: T = (1.00 atm * 22.4 L) / (1.00 mol * 0.0821 L*atm/mol*K)

Calculate the temperature

Now, we can calculate the temperature: T = (1.00 atm * 22.4 L) / (1.00 mol * 0.0821 L*atm/mol*K) ≈ 273 K The temperature of the helium gas in the 22.4-L vessel at a pressure of 1.00 atm is approximately 273 K.

Key concepts

Understanding Gas Laws

Gas laws are a fundamental part of chemistry and physics, describing the behavior of gases in various conditions. These laws are crucial for understanding how gases react to changes in temperature, volume, and pressure. The most famous gas laws include Boyle's Law, which relates pressure and volume at a constant temperature; Charles's Law, which relates volume and temperature at constant pressure; and Gay-Lussac's Law, relating pressure and temperature at constant volume. However, for exercises involving multiple changing conditions, the ideal gas law is employed.

The ideal gas law is expressed as the equation PV = nRT, combining the variables of pressure (P), volume (V), moles of gas (n), the ideal gas constant (R), and temperature (T). This holistic equation can be applied to predict how a given quantity of gas will behave when subject to varying pressures, volumes, and temperatures.

Molar Mass of Helium

The molar mass of an element is the mass of one mole of its atoms, usually expressed in grams per mole (g/mol). For helium, one of the lightest and simplest elements, the molar mass is 4.00 g/mol. This value is used to convert between the mass of helium and the number of moles of helium, a necessary step in calculations involving the ideal gas law.

Since helium is a noble gas, it is often used in gas law problems because it behaves very closely to an ideal gas, which is a hypothetical gas that perfectly follows the gas laws. The molar mass of helium is derived from its atomic weight, which is based on the average mass of all the isotopes of helium as they occur naturally.

Calculating Moles

The concept of moles is central to chemistry. A mole is a unit that represents 6.022 x 10^23 particles (Avogadro's number) of a substance. Calculating moles allows chemists to count particles by weighing them. To calculate the moles of a gas, like helium in our exercise, you divide the mass of the gas by its molar mass.

For example, in the step-by-step solution provided, the calculation was: = 4.00 g / 4.00 g/mol = 1.00 mol.
This straightforward calculation is integral to using the ideal gas law effectively because it allows for the quantification of how much gas is present and subsequently how it may behave under specific conditions.

Temperature Calculation

In the context of the ideal gas law, temperature calculation is essential. Temperature must always be expressed in Kelvin in these calculations, as this is an absolute temperature scale starting from absolute zero, where particles have minimum thermal motion.

To convert Celsius to Kelvin, one adds 273.15 to the Celsius temperature. In the given exercise, solving for temperature using the ideal gas law required re-arranging the formula to solve for T (temperature), resulting in T = PV / (nR). Upon substitution of known values, the temperature can be calculated. In our example, the temperature value for helium gas was found to be approximately 273 K, which is roughly equivalent to 0°C, the freezing point of water—a handy point of reference for understanding real-world gas behaviors.