Question

For each of the following sets of volume/temperature data, calculate the missing quantity. Assume that the pressure and the amount of gas remain constant. a. \(V=100 .\) mL at \(74^{\circ} \mathrm{C} ; V=?\) at \(-74^{\circ} \mathrm{C}\) b. \(V=500 .\) mL at \(100^{\circ} \mathrm{C} ; V=600 . \mathrm{mL}\) at \(?^{\circ} \mathrm{C}\) c. \(V=10,000 \mathrm{L}\) at \(25^{\circ} \mathrm{C} ; V=?\) at \(\overrightarrow{0} \mathrm{K}\)

Short answer

a. \(V_2 ≈ 57.35 \text{ mL}\) b. \(T_2 = 175°C\) c. Cannot be calculated using Charles's Law at 0K.

Step-by-step solution

Convert the given temperatures to Kelvin

Convert both temperatures to Kelvin: \[ T_1 = 74°C + 273 = 347K \] \[ T_2 = -74°C + 273 = 199K \]

Use Charles's Law to find the missing volume

Plug the values into Charles's Law equation and solve for \(V_2\): \[ \frac{100 \text{ mL}}{347 \text{ K}} = \frac{V_2}{199 \text{ K}} \] \[ V_2 = \frac{100 \text{ mL} \times 199 \text{ K}}{347 \text{ K}} \] \[ V_2 ≈ 57.35 \text{ mL} \] #b. Calculate the missing temperature#

Convert the given temperature to Kelvin

Convert the initial temperature to Kelvin: \[ T_1 = 100°C + 273 = 373K \]

Use Charles's Law to find the missing temperature

Plug the values into Charles's Law equation and solve for \(T_2\): \[ \frac{500 \text{ mL}}{373 \text{ K}} = \frac{600 \text{ mL}}{T_2} \] \[ T_2 = \frac{600 \text{ mL} \times 373 \text{ K}}{500 \text{ mL}} \] \[ T_2 ≈ 448 \text{ K} \] Convert the final temperature back to Celsius: \[ T_2 = 448 \text{ K} - 273 = 175°C \] #c. Calculate the missing volume#

Convert the given temperature to Kelvin

Convert the initial temperature to Kelvin: \[ T_1 = 25°C + 273 = 298K \] Since the final temperature is given in Kelvin already, \[ T_2 = 0K \]

Use Charles's Law to find the missing volume

Plug the values into Charles's Law equation and solve for \(V_2\): \[ \frac{10,000 \text{ L}}{298 \text{ K}} = \frac{V_2}{0 \text{ K}} \] Charles's Law is undefined when the temperature is absolute 0 (0K). Thus, we cannot calculate the volume at 0K using Charles's Law in this case. It is important to mention that a real gas cannot reach 0K, as it would violate the laws of thermodynamics.

Key concepts

Temperature Conversion

Temperature conversion is crucial when dealing with gas calculations, such as those involving Charles's Law. In most gas law problems, the temperatures are initially given in Celsius and need to be converted into Kelvin. This conversion is simple and follows a straightforward formula:

• Add 273 to the given Celsius temperature to convert it to Kelvin.

For example, to convert 74°C to Kelvin, you simply do:\[ T = 74 + 273 = 347 \text{ K} \]Converting back from Kelvin to Celsius requires subtracting 273:

• For instance, if you have a temperature of 448 K and wish to express it in Celsius, subtract 273 to obtain:

\[ T = 448 - 273 = 175°C \]Remember, Kelvin is the preferred unit for thermodynamic studies because it starts from absolute zero – the point where theoretically, particles have minimum thermal motion.

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry and physics that helps predict the behavior of gases under different conditions. It is typically expressed as:\[ PV = nRT \]where:

• \( P \) is the pressure,
• \( V \) is the volume,
• \( n \) is the number of moles of the gas,
• \( R \) is the ideal gas constant, and
• \( T \) is the temperature in Kelvin.

In the context of Charles's Law, which is a simpler version focusing on volume and temperature at a constant pressure and amount of gas, the relationship is expressed as:\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]where the volumes and temperatures at two different states are compared, keeping pressure and quantity constant. This allows us to solve for unknown pressures or temperatures, as seen in many gas law problems.Charles's Law shows how gas expands with temperature: if the pressure is kept constant, increasing the temperature results in an increase in volume and vice versa.

Thermodynamics

Thermodynamics is the study of the relationship between heat, work, temperature, and energy within physical systems. A critical aspect of thermodynamics involves understanding how energy transfers and transforms. One fundamental principle in thermodynamics is the absolute zero temperature, or 0 Kelvin, which represents the lower limit of temperatures. At this point, a gas would theoretically have no kinetic energy, meaning its particles would be motionless. However, reaching absolute zero is practically impossible for real gases due to the Third Law of Thermodynamics.

• This law states that as a system approaches absolute zero, all processes virtually cease.
• Thus, the entropy, or disorder, of a perfect crystal approaches a constant minimum.

In the exercise given, trying to calculate the volume at absolute zero highlights its theoretical nature — it's a reminder that this temperature is more a concept than a state gases can physically achieve under natural circumstances.