Question

A bacterial culture isolated from sewage produced \(51.5 \mathrm{~mL}\) of methane, \(\mathrm{CH}_{4}\), at \(33^{\circ} \mathrm{C}\) and \(752 \mathrm{mmHg}\). What is the volume of this methane at standard temperature and pressure \(\left(0^{\circ} \mathrm{C}, 760 \mathrm{mmHg}\right) ?\)

Short answer

The methane volume at STP is approximately 46.02 mL.

Step-by-step solution

Understand the Parameters

We are tasked with finding the volume of methane at standard temperature and pressure (STP), which is defined as 0°C and 760 mmHg. We start with the initial conditions: volume = 51.5 mL, temperature = 33°C, and pressure = 752 mmHg.

Convert Celsius to Kelvin

To use the Ideal Gas Law, temperatures must be in Kelvin. Convert 33°C to Kelvin by adding 273.15.\[ T_1 = 33 + 273.15 = 306.15 \text{ K} \]Standard temperature at STP is 0°C, which converts to:\[ T_2 = 0 + 273.15 = 273.15 \text{ K} \]

Use the Combined Gas Law

The Combined Gas Law relates initial and final states of a gas:\[ \frac{P_1 \cdot V_1}{T_1} = \frac{P_2 \cdot V_2}{T_2} \]Substituting in the known values (where all pressures are in mmHg and temperatures in Kelvin):\[ \frac{752 \cdot 51.5}{306.15} = \frac{760 \cdot V_2}{273.15} \]

Solve for the Final Volume

Rearrange the equation to solve for \( V_2 \):\[ V_2 = \frac{752 \cdot 51.5 \cdot 273.15}{306.15 \cdot 760} \]Calculate:\[ V_2 = \frac{10540.8 \cdot 273.15}{232674} \approx 46.02 \text{ mL} \]

Interpret the Result

After solving the equation, we find that the volume of the methane at standard temperature and pressure is approximately 46.02 mL.

Key concepts

Combined Gas Law

The Combined Gas Law is a useful equation that combines three other gas laws: Boyle's Law, Charles's Law, and Gay-Lussac's Law. These laws describe how gas behaves under certain conditions, such as changes in pressure, volume, and temperature. The Combined Gas Law allows us to calculate the changes in a gas sample when conditions change by holding one of the properties constant. The formula for the Combined Gas Law is:

• \( \frac{P_1 \cdot V_1}{T_1} = \frac{P_2 \cdot V_2}{T_2} \)

Here, \( P \) stands for pressure, \( V \) is volume, \( T \) is temperature in Kelvin, and the subscripts 1 and 2 refer to the initial and final states of the gas, respectively.
To use this law effectively, remember to convert all temperatures to Kelvin as gas law equations rely on absolute temperature. The Combined Gas Law makes it simple to predict how gas will respond to shifts in its environment, which is why it's widely used in chemistry.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) is a standardized set of conditions for experimental measurements within the context of gas behavior. It’s defined as a temperature of 0°C (or 273.15 K) and a pressure of 760 mmHg (which is equivalent to 1 atm or 101.3 kPa).
STP serves as a reference point, which is important when comparing the characteristics of gases. By agreeing on this standard, scientists ensure that their findings are comparable, even when recorded in different regions or labs. Moreover, many calculations in chemistry, including those involving the Ideal Gas Law and Combined Gas Law, simplify significantly at these conditions.
This is why, in the original exercise, knowing the methane volume at STP was crucial. It allows predictions and comparisons against other gases also measured under STP.

Gas Volume Calculations

Gas volume calculations involve determining how much space a gas occupies under particular conditions. This is important in chemistry and physics because gases are often involved in reactions or as part of atmospheric studies.
In the given exercise, calculating the volume of methane produced at different conditions used the Combined Gas Law. Initially, the volume observed was at 33°C and 752 mmHg. To find out what the volume would be at STP, we first convert the temperature from Celsius to Kelvin using the formula:

• \( T(K) = T(°C) + 273.15 \)

Next, we apply the Combined Gas Law to calculate the new volume when it is shifted to the STP conditions.
Calculating these changes accurately requires careful adherence to units and conversions, especially with temperature and pressure. Understanding gas volume calculations is essential for anyone studying the properties and reactions of gases.