Question

A given mass of gas occupies a volume of \(435 \mathrm{~mL}\) at \(28^{\circ} \mathrm{C}\) and \(740 \mathrm{mmHg}\). What will be the new volume at STP?

Short answer

The new volume at STP is approximately 384.3 mL.

Step-by-step solution

Understand Standard Temperature and Pressure (STP)

At standard temperature and pressure (STP), the temperature is defined as 0°C (or 273 K) and the pressure is 760 mmHg (or 1 atm). We need to convert our conditions to STP to find the new volume.

Convert the Initial Temperature to Kelvin

Temperature in Celsius must be converted into Kelvin by adding 273 to the Celsius value. For our problem, the temperature is 28°C.\[T_1 = 28 + 273 = 301 \, \text{K}\]

Apply the Combined Gas Law

Since the number of moles of gas and the gas constant are constant, we can use the combined gas law: \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \), where \(P\) is pressure, \(V\) is volume, and \(T\) is temperature.Given:- \(P_1 = 740 \, \text{mmHg}\)- \(V_1 = 435 \, \text{mL}\)- \(T_1 = 301 \, \text{K}\)- \(P_2 = 760 \, \text{mmHg}\)- \(T_2 = 273 \, \text{K}\)

Rearrange the Combined Gas Law for V2

Rearrange the formula to solve for \(V_2\):\[ V_2 = \frac{P_1 V_1 T_2}{T_1 P_2} \]

Substitute the Values into the Equation

Substitute the known values into the equation:\[V_2 = \frac{740 \, \text{mmHg} \times 435 \, \text{mL} \times 273 \, \text{K}}{301 \, \text{K} \times 760 \, \text{mmHg}}\]

Calculate V2

Perform the calculations:\[V_2 = \frac{740 \times 435 \times 273}{301 \times 760} = \frac{87,993,900}{228,760} \approx 384.3 \, \text{mL}\]Thus, the new volume \(V_2\) at STP is approximately 384.3 mL.

Key concepts

Standard Temperature and Pressure (STP)

When dealing with gases, scientists often refer to Standard Temperature and Pressure, known as STP. This is a standard set of conditions to make sure results are consistent and comparable.
At STP, the temperature is set at 0°C or 273 Kelvin, and the pressure is 760 mmHg or 1 atmosphere (atm).
These numbers are important because they provide a baseline for calculations that involve gases, allowing for accurate predictions and comparisons. Understanding STP is crucial for solving problems related to gas behavior under differing conditions. By knowing the conditions at STP, we can predict how gas volume will change when adjusted to this standard baseline. This is the key to converting and comparing measurements to understand gas behavior correctly.

Kelvin Temperature Conversion

Temperature is a vital factor in gas calculations. The Kelvin scale is commonly used because it avoids negative values, which complicate calculations.
To convert from Celsius to Kelvin, a simple addition is performed: 273. This is because 0 Kelvin, also known as absolute zero, is equivalent to -273°C.
For instance, an initial temperature of 28°C becomes 301 Kelvin, as calculated by adding 273 to 28. Using Kelvin is essential for the Combined Gas Law and other thermodynamic equations as it provides a linear scale. This linearity ensures that direct relationships between temperature and other gas variables, such as volume and pressure, can be utilized effectively in calculations.

Gas Volume Calculation

Gas volume calculation relies heavily on the principles outlined in the Combined Gas Law. This law states that the product of pressure and volume, divided by temperature, remains constant as long as the amount of gas remains unchanged.The equation used is \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \), where:

• \(P\) denotes pressure
• \(V\) stands for volume
• \(T\) represents temperature

By rearranging this equation, we can solve for an unknown variable, typically the new volume or \(V_2\). For example, if changing conditions from 740 mmHg and 301 Kelvin to STP values, the new volume can be found as follows:
\( V_2 = \frac{P_1 V_1 T_2}{T_1 P_2} \). After substituting the known values, calculate \(V_2\) to find the new volume at STP.
This process helps us understand how gases behave under different pressures and temperatures, facilitating practical applications such as predicting how a gas will respond to environmental changes.