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At constant pressure a certain gas at \(0^{\circ} \mathrm{C}\), was cooled until its volume was reduced to half. The temperature at this stage (1) \(-130.5^{\circ} \mathrm{C}\) (2) \(-140.0^{\circ} \mathrm{C}\) (3) \(-136.5^{\circ} \mathrm{C}\) (4) \(-120^{\circ} \mathrm{C}\)

Short Answer

Expert verified
-136.5°C

Step by step solution

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01

- Understand Given Conditions

The problem states that a gas is initially at \(0^{\text{°}} \text{C}\) and is cooled until its volume is reduced to half. The process occurs at constant pressure.
02

- Convert Initial Temperature to Kelvin

Convert the initial temperature from Celsius to Kelvin. The temperature at \(0^{\text{°}} \text{C}\) is \(273.15 \text{K}\): \[ T_1 = 0^{\text{°}} \text{C} + 273.15 = 273.15 \text{K} \]
03

- Apply Charles's Law

According to Charles's Law, \( V_1 / T_1 = V_2 / T_2 \). Here \( V_2 = V_1 / 2 \) since the volume is reduced to half. Substituting these values, we get \[ \frac{V_1}{273.15} = \frac{V_1/2}{T_2} \] which simplifies to \[ T_2 = \frac{273.15}{2} = 136.575 \text{K} \]
04

- Convert Final Temperature to Celsius

Convert the final temperature back to Celsius: \[ T_2 = 136.575 \text{K} - 273.15 = -136.575^{\text{°}} \text{C} \] The closest option provided is \( -136.5^{\text{°}} \text{C} \).

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gas Laws
Gas laws are a set of rules that describe how gases behave under different conditions, such as pressure, volume, and temperature. One of the key gas laws is Charles's Law. Charles's Law states that at constant pressure, the volume of a gas is directly proportional to its absolute temperature.

This means that if the temperature of the gas increases, the volume will also increase, provided the pressure is constant. Conversely, if the temperature decreases, the volume will decrease. This relationship can be expressed mathematically as
\overset{\text{direct proportionality}}{\text{V_1 / T_1 = V_2 / T_2}}

Other primary gas laws include:
  • Boyle's Law: At constant temperature, the volume of a gas is inversely proportional to its pressure\( \text{P_1 V_1 = P_2 V_2} \)

  • Gay-Lussac's Law: At constant volume, the pressure of a gas is directly proportional to its absolute temperature\( \text{P_1 / T_1 = P_2 / T_2} \)

  • Avogadro's Law: At constant temperature and pressure, the volume of a gas is directly proportional to the number of molecules\( \text{V_1 / n_1 = V_2 / n_2} \)



Understanding these laws helps predict and explain the behavior of gases in different scenarios.
Temperature Conversion
In gas law calculations, it's crucial to always work with the Kelvin temperature scale. Kelvin is the SI unit for temperature and starts from absolute zero, which is -273.15°C. This scale ensures that temperature values are always positive, which is essential for calculations in gas laws.

To convert Celsius to Kelvin, simply add 273.15:

\[T_{(K)} = T_{(°C)} + 273.15\]

For example, if a gas is initially at 0°C, you convert it to Kelvin as follows:

\[T_{(K)} = 0 + 273.15 = 273.15 \text{K}\]

To convert from Kelvin back to Celsius, subtract 273.15:

\[T_{(°C)} = T_{(K)} - 273.15\]

For instance, if the temperature in Kelvin is 136.575K, it converts to Celsius like this:

\[T_{(°C)} = 136.575 - 273.15 = -136.575 \text{°C}\]

Temperature conversion is an essential step in solving problems involving Charles's Law and other gas laws. Always double-check your conversions to ensure accuracy in your final calculations.
Volume-Temperature Relationship
The volume-temperature relationship in gases is articulated by Charles's Law. This law states that at constant pressure, the volume and absolute temperature of a gas are directly proportional. This is why the equation \(V_1 / T_1 = V_2 / T_2\) is used to describe this relationship.

Here’s a step-by-step example of how Charles's Law is applied:
  • Step 1: Identify the initial and final conditions for volume and temperature. For example, a gas initially at 0°C and a certain volume (<\(V_1\)) is cooled to reduce its volume by half (<\(V_2 = V_1 / 2\)).

  • Step 2: Convert the initial temperature from Celsius to Kelvin:

    \(T_1 = 0^{\text{°}} \text{C} + 273.15 = 273.15 \text{K}\)

  • Step 3: Apply Charles's Law. Substitute the known values into the equation:
    \(\frac{V_1}{273.15} = \frac{V_1/2}{T_2}\)

  • Step 4: Solve for the final temperature in Kelvin (\T_2):

    \(T_2 = 273.15 / 2 = 136.575 \text{K}\)

  • Step 5: Convert the final temperature back to Celsius:

    \(T_2 = 136.575 \text{K} - 273.15 = -136.575^{\text{°}} \text{C}\)



This process illustrates how changes in temperature affect the volume of a gas, as long as the pressure remains constant. Charles's Law helps us predict this behavior accurately.

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