Problem 28 - Chemistry Matter and Change

Question

What is the pressure, in atmospheres, of a 0.108-mol sample of helium gas at a temperature of 20.0°C if its volume is 0.505 L?

Step-by-Step Solution

Answer

The pressure of the 0.108-mol sample of helium gas at 20.0°C and a volume of 0.505 L is approximately \(4.99 \: \text{atm}\).

Step 1: Convert the temperature to Kelvin

To convert the temperature from Celsius to Kelvin, use the formula: T(K) = T(°C) + 273.15 T(K) = 20.0°C + 273.15 = 293.15 K The temperature, in Kelvin, is 293.15 K.

Step 2: Identify the given values and the ideal gas constant

We are given: - n (number of moles of helium gas) = 0.108 mol - V (volume of the gas) = 0.505 L - T (temperature in Kelvin) = 293.15 K - R (ideal gas constant) = 0.0821 atm•L•mol⁻¹•K⁻¹

Step 3: Plug the given values into the ideal gas law equation

Using the ideal gas law equation PV = nRT, we can plug in the given values: P × 0.505 L = 0.108 mol × 0.0821 atm•L•mol⁻¹•K⁻¹ × 293.15 K

Step 4: Solve for pressure (P)

To solve for pressure (P), divide both sides of the equation by the volume (0.505 L): P = (0.108 mol × 0.0821 atm•L•mol⁻¹•K⁻¹ × 293.15 K) / 0.505 L Now, perform the calculations: P = (0.108 × 0.0821 × 293.15) / 0.505 = 4.9856 atm

Step 5: Round your answer

Round the pressure to an appropriate number of significant figures: P = 4.99 atm (rounded to 2 decimal places for three significant figures) The pressure of the 0.108-mol sample of helium gas at 20.0°C and a volume of 0.505 L is approximately 4.99 atm.

Key Concepts

Understanding Gas Pressure Calculation

The concept of gas pressure is fundamental in understanding how gases behave. In the context of the ideal gas law, pressure represents one of the variables that define the state of a gas. To calculate the gas pressure, one can use the ideal gas law, which states that for a certain amount of gas at a constant temperature, the pressure is inversely proportional to the volume.

The ideal gas law is represented by the equation PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. By rearranging the equation for P, we get P = (nRT)/V. This equation implies that if we know the amount of gas (in moles), the temperature in Kelvin, and the volume of the gas, we can calculate its pressure.

In our exercise example, after plugging the known values into the ideal gas law equation, we solve for P to find the pressure in atmospheres.

Mastering Temperature Conversion

Temperature conversion from Celsius to Kelvin is a vital step in gas calculations, as the ideal gas law requires temperature to be in Kelvin. Kelvin is the SI unit for temperature, and it is used in scientific measurements because it starts at absolute zero, unlike Celsius or Fahrenheit scales. The conversion formula is T(K) = T(°C) + 273.15. This calculation ensures that all temperature measurements are on the same scale for accurate results.

In practice, if you're given a temperature like 20.0°C, you convert it to Kelvin by simply adding 273.15. Therefore, 20.0°C becomes 293.15 K. Remember, there are no negative temperatures in Kelvin, which makes it perfect for calculations involving physical and chemical processes like gas reactions.

Explaining Moles and Volume Relationship

The relationship between moles and volume of a gas is described by Avogadro's law, which states that equal volumes of gases at the same temperature and pressure contain an equal number of moles. This is useful when working with the ideal gas law.

In simple terms, the volume of a gas is directly proportional to the number of moles when temperature and pressure are held constant. The volume occupied by one mole of gas at standard temperature and pressure (STP - 0°C and 1 atm) is 22.4 liters. However, when the conditions are not at STP, we can calculate the volume using the ideal gas law.

In the problem we faced, the number of moles was given, and the volume was known. This allows us to use these quantities along with the ideal gas constant and temperature to calculate the pressure. As demonstrated, the moles and volume relationship plays a critical role in fully understanding gas behavior and is an important concept within the ideal gas law framework.