Problem 165 - Chemistry The Science in Context

Question

A sample of \(11.4 \mathrm{L}\) of an ideal gas at \(25.0^{\circ} \mathrm{C}\) and 735 torr is compressed and heated so that the volume is \(7.9 \mathrm{L}\) and the temperature is \(72.0^{\circ} \mathrm{C} .\) What is the pressure in the container?

Step-by-Step Solution

Answer

Answer: The final pressure in the container is 1.45 atm.

Step 1: Convert the given temperatures to Kelvin

To work with the Ideal Gas Law, we need to convert the given temperatures from Celsius to Kelvin. The conversion is done using the following relation: T(K) = T(°C) + 273.15 Initial temperature, T1 = 25.0°C + 273.15 = 298.15 K Final temperature, T2 = 72.0°C + 273.15 = 345.15 K

Step 2: Convert the given pressure from torr to atm

We also need to convert the given pressure from torr to atm for consistency with the Ideal Gas Law formula. The conversion factor is 1 atm = 760 torr. Initial pressure, P1 = 735 torr * (1 atm / 760 torr) = 0.967 atm

Step 3: Use the combined gas law equation to find the final pressure

Now, we can plug in the values into the combined gas law equation: P1V1/T1 = P2V2/T2 0.967 atm * 11.4 L / 298.15 K = P2 * 7.9 L / 345.15 K Now, we just need to solve for P2 (final pressure). P2 = 0.967 atm * 11.4 L / 298.15 K * 345.15 K / 7.9 L P2 = 1.45 atm The final pressure in the container is 1.45 atm.

Key Concepts

Combined Gas Law Equation

The combined gas law is a fundamental equation that blends the three gas laws: Boyle's Law, Charles's Law, and Gay-Lussac's Law. It is useful when dealing with scenarios where pressure, volume, and temperature of a gas are changing simultaneously.

The combined gas law can be represented as:
\[ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \]
where:

• \(P_1\) and \(P_2\) are the initial and final pressures,
• \(V_1\) and \(V_2\) are the initial and final volumes, and
• \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin.

In a practical sense, this equation enables us to predict how a gas will react to various changes in conditions. For instance, if a gas is compressed, the volume decreases and, if temperature is constant, the pressure increases according to Boyle's Law. However, if we allow the gas to heat up, the temperature rise can offset the pressure increase as per Charles's Law. The combined gas law equation ties these individual relationships together, offering a comprehensive description of the gas behavior.

Gas Pressure Conversion

When working with gasses, pressure can be measured in multiple units such as atmospheres (atm), torr, or Pascals (Pa). The Ideal Gas Law requires a consistent set of units, typically atmospheres for pressure.

Conversion between units is straightforward but essential for accurate calculations. The conversion factor between torr and atmosphere is that 1 atm equals 760 torr.To convert from torr to atm, you would divide the pressure in torr by 760. Similarly, to convert from atm to torr, you multiply by 760.

Mathematically, the conversion formulas are:

• \( P(atm) = \frac{P(torr)}{760} \)
• \( P(torr) = P(atm) \times 760 \)

Understanding and correctly applying these conversions ensures that the pressure values you work with are consistent, which is critical in solving problems using the Ideal Gas Law or the combined gas law equation.

Kelvin Temperature Scale

The Kelvin temperature scale is an absolute temperature scale, utilizing the same increment size as the Celsius scale, but with its zero point at absolute zero, the theoretical lowest temperature possible. Absolute zero equates to -273.15°C.

For gas law calculations, temperatures must be converted to the Kelvin scale because gas volumes are directly proportional to temperature in Kelvins and pressures are due to particle collisions, which increase with temperature. The Kelvin scale ensures that these relationships remain linear and proportional.

To convert from Celsius to Kelvin, the formula is:

• \(T(K) = T(°C) + 273.15\)

Notably, there are no negative temperatures on the Kelvin scale. This property makes it particularly useful for scientific calculations, as it simplifies the mathematics when dealing with temperature and energy. In the study of Ideal Gas Law, the use of Kelvin temperature is paramount to avoiding unphysical results such as negative volumes or pressures.

Gas Volume and Temperature Relationship

The relationship between gas volume and temperature is described by Charles's Law, which states that, at a constant pressure, the volume of a gas is directly proportional to its temperature in Kelvins. This means that if the temperature increases, the volume of the gas increases as well, if pressure remains unchanged.

The mathematical representation of Charles's Law is:

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

where \(V_1\) and \(V_2\) are the initial and final volumes, and \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin. To fully grasp this concept, imagine inflating a balloon. As it is heated, the air inside expands and the volume of the balloon increases. Conversely, cooling the balloon would make the air contract and decrease its volume.

Understanding this relationship is crucial in various scientific fields, such as meteorology and aeronautics, and is foundational in gas law problems. It’s a concept that is not only theoretically important but also practically observable in everyday life.