Problem 33 - University Physics with Modern Physics

Question

We have two equal-size boxes, \(A\) and \(B\) . Each box contains gas that behaves as an ideal gas. We insert a thermometer into each box and find that the gas in box \(A\) is at a temperature of \(50^{\circ} \mathrm{C}\) while the gas in box \(B\) is at \(10^{\circ} \mathrm{C}\) . This is all we know about the gas in the boxes. Which of the following statements must be true? Which could be true? (a) The pressure in \(A\) is higher than in \(B\) . (b) There are more molecules in \(A\) than in \(B\) . (c) \(A\) and \(B\) do not contain the same type of gas. (d) The molecules in \(A\) have more average kinetic energy per molecule than those in \(B\) . (e) The molecules in \(A\) are moving faster than those in \(B .\) Explain the reasoning behind your answers.

Step-by-Step Solution

Answer

Statements (d) and (e) are definitely true; statement (a) could be true.

Step 1: Understanding Ideal Gas Behavior

An ideal gas follows the equation \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the amount of substance, \(R\) is the gas constant, and \(T\) is temperature in Kelvin. The temperature affects the energy and speed of molecules.

Step 2: Converting Temperatures to Kelvin

Convert the temperatures from Celsius to Kelvin by adding 273.15. So, the temperatures are \(T_A = 323.15 \, \text{K}\) and \(T_B = 283.15 \, \text{K}\).

Step 3: Analyzing Pressure Difference (a)

According to the ideal gas law, pressure is directly proportional to temperature if volume and moles of gas are constant. Therefore, since \(T_A > T_B\), pressure \(P_A\) could be higher than \(P_B\), but it depends on other factors like number of molecules and nature of gases.

Step 4: Analyzing Number of Molecules (b)

The number of molecules is determined by the amount of substance \(n\). With no information about \(n\), we cannot determine if there are more molecules in \(A\) than \(B\).

Step 5: Analyzing Gas Types (c)

Different gases would have different molecular structures and masses, but without additional information, we cannot determine if the gases are different solely based on temperature.

Step 6: Analyzing Average Kinetic Energy (d)

The average kinetic energy of a gas molecule is given by \( \frac{3}{2}kT \), where \(k\) is Boltzmann's constant. Since \(T_A > T_B\), molecules in \(A\) have more average kinetic energy than those in \(B\). This is true.

Step 7: Analyzing Molecular Speed (e)

For ideal gases, the root mean square speed is proportional to the square root of the temperature, \( \text{v}_{\text{rms}} = \sqrt{\frac{3kT}{m}} \). Since \(T_A > T_B\), molecules in \(A\) move faster than those in \(B\), making this statement true.

Key Concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. It helps us understand how energy is transferred and transformed. One key concept in thermodynamics is the behavior of ideal gases, which follow the ideal gas law:

• The ideal gas law is expressed as \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is the absolute temperature in Kelvin.
• It demonstrates the direct proportionality between pressure and temperature, suggesting that for a constant volume and amount of gas, as the temperature rises, the pressure also increases.
• This principle is crucial for understanding that high temperature in a gas can lead to higher pressure, as seen in many practical thermodynamic systems like engines and refrigerators.

Understanding these concepts allows us to predict and analyze the behavior of gases under different conditions of pressure, volume, and temperature, which is critical in many scientific and engineering applications.

Kinetic Theory of Gases

The kinetic theory of gases provides a microscopic explanation of thermodynamics by considering the motion and energy of individual gas molecules.

• According to this theory, gas molecules are in constant random motion, colliding with one another and with the walls of their container. These collisions result in gas pressure.
• Temperature, in this context, is a measure of the average kinetic energy of the molecules in the gas. Therefore, higher temperatures mean higher average speeds and more energetic particle movement.
• The average kinetic energy per molecule is captured in the expression \(\frac{3}{2}kT\), where \(k\) is the Boltzmann constant and \(T\) is the temperature in Kelvin.

In our exercise, the gas in box A at a higher temperature has molecules with more average kinetic energy than those in box B. This increased kinetic energy also translates to faster-moving molecules, which reinforces the concept that temperature is directly linked to molecular activity and speed.

Temperature Conversion

Temperature conversion, particularly from Celsius to Kelvin, is an essential step in many physics calculations, including those involving the ideal gas law.

• Kelvin is the standard unit of temperature in science because it begins at absolute zero, the point where all molecular motion theoretically stops.
• To convert from Celsius to Kelvin, simply add 273.15 to the Celsius temperature. So for box A, \(50^{\circ}C = 323.15 \, K\) and for box B, \(10^{\circ}C = 283.15 \, K\).
• Using Kelvin allows for direct correlation with energy and kinetic theories since Kelvin directly links to molecular kinetic energy and movement.

Always ensure temperatures used in scientific calculations are in Kelvin to get accurate and meaningful results, enabling proper application of formulas in thermodynamics and gas behavior.