Question

500 liters of a gas at \(27^{\circ} \mathrm{C}\) and 700 torr would occupy what volume at STP?

Short answer

The gas would occupy a volume of \(452.3 \, \mathrm{L}\) at STP.

Step-by-step solution

Convert initial temperature to Kelvin

To convert the initial temperature from Celsius to Kelvin, we simply add 273.15 to the given temperature: Initial temperature in Kelvin = \(27^{\circ} \mathrm{C} + 273.15 \, \mathrm{K} = 300.15 \, \mathrm{K}\)

Convert initial pressure to atm

To convert the given pressure from torr to atm, we use the conversion factor: \(1 \, \mathrm{atm} = 760 \, \mathrm{torr}\) Initial pressure in atm = \(\frac{700 \, \mathrm{torr}}{760 \, \mathrm{torr/atm}} = 0.921 \, \mathrm{atm}\)

Apply the ideal gas law to find the number of moles

Using the ideal gas law formula (\(PV = nRT\)), we can find the number of moles (n) of the gas: Rearrange the formula to find n: \(n = \frac{PV}{RT}\) Plugging in the values: \(n = \frac{0.921 \, \mathrm{atm} \times 500 \, \mathrm{L}}{0.0821 \, \mathrm{L \cdot atm \cdot K^{-1} \cdot mol^{-1}} \times 300.15 \, \mathrm{K}} = 20.0 \, \mathrm{moles}\)

Calculate the volume at STP

Now that we know the number of moles (n), we can use the ideal gas law formula again, this time at STP conditions: STP pressure (P) = 1 atm STP temperature (T) = 273.15 K Rearrange the formula to find the volume (V) at STP: \(V = \frac{nRT}{P}\) Plugging in the values: \(V_\text{STP} = \frac{20.0 \, \mathrm{moles} \times 0.0821 \, \mathrm{L \cdot atm \cdot K^{-1} \cdot mol^{-1}} \times 273.15 \, \mathrm{K}}{1 \, \mathrm{atm}} = 452.3 \, \mathrm{L}\) So, the gas would occupy a volume of 452.3 liters at STP.

Key concepts

Converting Celsius to Kelvin

Understanding temperature conversions is crucial in chemistry, especially when dealing with gas laws. Celsius and Kelvin are two temperature scales that are interconnected. To convert Celsius to Kelvin, which is necessary for calculations like those involving the ideal gas law, one simply adds 273.15 to the Celsius value. This is because the zero point in the Kelvin scale (absolute zero) is equivalent to -273.15 degrees Celsius.

For instance, in the practice problem provided, the temperature in Celsius is given as 27°C. Using the conversion formula: Initial temperature in Kelvin = 27°C + 273.15 = 300.15 K.This step ensures that the temperature is in the correct unit for subsequent calculations involving the ideal gas law. Keeping track of units and using correct conversions are foundational skills for any student studying chemistry.

Converting Torr to ATM

Pressure has various units, and often, conversions are required to use the ideal gas law, which typically uses atmospheric pressure (atm) as the standard unit. One common pressure unit used is the torr. To convert torr to atm, we use the relationship where 1 atm is equal to 760 torr.

So, to convert a pressure reading from torr to atm, we divide the number of torr by 760. For example, if we have a pressure of 700 torr, we convert it as follows:Initial pressure in atm = 700 torr / 760 torr/atm ≈ 0.921 atm.It's a simple yet essential conversion that ensures compatibility with the ideal gas law formula.

STP Conditions

STP stands for standard temperature and pressure, which are defined conditions used as reference points in chemistry, especially when discussing gases. The standard temperature is defined as 0°C, which is equivalent to 273.15 K, and the standard pressure is 1 atm. These conditions are essential for comparing the behaviors of gases under different scenarios.

By using STP, chemists can confidently communicate and compare findings, knowing that the conditions are universally understood to be at these specific values. When solving problems involving gas volume conversions, understanding and using STP conditions correctly are crucial for obtaining accurate and comparable results.

Gas Volume Conversion

Converting the volume of a gas from one set of conditions to another is a common exercise when studying gases. The ideal gas law, represented by the equation PV = nRT,where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature, provides a reliable mathematical framework for such conversions.

In the example at hand, once we’ve determined the number of moles present, we can calculate the gas volume under STP conditions by rearranging and applying the ideal gas law. We use the known values of the universal gas constant, the standard temperature (273.15 K), and the standard pressure (1 atm) to find the new volume: V_STP = (nRT)/P = (20.0 moles * 0.0821 L.atm.K^-1.mol^-1 * 273.15 K) / 1 atm ≈ 452.3 L.

This illustrates the direct relationship between volume and both temperature and pressure under the watchful eye of the ideal gas law.