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Five moles of an ideal monatomic gas with an initial temperature of 127\(^\circ\)C expand and, in the process, absorb 1500 J of heat and do 2100 J of work. What is the final temperature of the gas?

Short Answer

Expert verified
The final temperature of the gas is 390.5 K.

Step by step solution

01

Convert Initial Temperature to Kelvin

The initial temperature of the gas is given as 127\(^\circ\)C. To convert this to Kelvin, we add 273.15. Thus, \(T_i = 127 + 273.15 = 400.15\,\text{K}\).
02

Apply the First Law of Thermodynamics

The first law of thermodynamics is given by \( \Delta U = Q - W \), where \( \Delta U \) is the change in internal energy, \( Q \) is the heat absorbed, and \( W \) is the work done by the system. Here, \( Q = 1500\, J \) and \( W = 2100\, J \). Thus, \( \Delta U = 1500 - 2100 = -600\, J \).
03

Relate Internal Energy Change to Temperature Change

For an ideal monatomic gas, the change in internal energy can also be expressed using \( \Delta U = \frac{3}{2}nR\Delta T \), where \( n \) is the number of moles and \( R \) is the ideal gas constant \( 8.314\, J/(mol\cdot K) \).
04

Solve for Change in Temperature \( \Delta T \)

Using \( \Delta U = \frac{3}{2}nR\Delta T \), we have: \[ -600 = \frac{3}{2} \times 5 \times 8.314 \times \Delta T \]Solve for \( \Delta T \):\[ \Delta T = \frac{-600}{\frac{3}{2} \times 5 \times 8.314} \]\[ \Delta T \approx -9.65\, K \]
05

Calculate the Final Temperature

Given \( T_i = 400.15\, K \) and \( \Delta T = -9.65\, K \), the final temperature \( T_f = T_i + \Delta T = 400.15 - 9.65 = 390.5\, K \).
06

Convert Final Temperature to Celsius (if needed)

To convert the final temperature from Kelvin to Celsius: \[ T_f(\degree C) = 390.5 - 273.15 = 117.35\, \degree C \]

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Key Concepts

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

first law of thermodynamics
The first law of thermodynamics is a fundamental principle that helps us understand energy changes within a system. It's closely related to the conservation of energy. The law states: \( \Delta U = Q - W \), where:
  • \( \Delta U \) is the change in internal energy of the system.
  • \( Q \) is the heat transferred into the system (positive if absorbed).
  • \( W \) is the work done by the system on its surroundings (positive if the system does work).
In this exercise, understanding how energy is shared between work and heat is crucial to find the change in internal energy. When the system absorbs 1500 J of heat and does 2100 J of work, it indicates energy is leaving the system, resulting in a negative internal energy change.
internal energy
Internal energy is associated with the microscopic components of a system, like molecules and atoms in a gas. For an ideal monatomic gas, internal energy depends solely on its temperature and is given by: \( U = \frac{3}{2} nRT \).
This equation links internal energy directly to the amount of heat added and the work performed.
  • \( n \) is the number of moles.
  • \( R \) is the ideal gas constant, \( 8.314\, J/(mol\cdot K) \).
  • \( T \) is the absolute temperature in Kelvin.
As energy is related to the temperature in an ideal monatomic gas, any change in internal energy \( \Delta U \) will result in a temperature change \( \Delta T \), calculated using \( \Delta U = \frac{3}{2}nR\Delta T \). This relationship helps solve the problem by revealing the decrease in internal temperature based on the energy exchange.
temperature conversion
Converting temperature from Celsius (°C) to Kelvin (K) is often necessary in thermodynamics. This conversion ensures that calculations align with related formulas that require absolute temperatures in Kelvin.
The conversion formula is straightforward: \( T(K) = T(°C) + 273.15 \).
For this exercise:
  • The initial temperature is 127°C. Add 273.15 to obtain 400.15 K.
  • After calculations, the final temperature of 390.5 K can be converted back to Celsius for practical understanding if needed, using \( T(°C) = T(K) - 273.15 \).
  • This results in a final temperature of 117.35°C.
Always remember, Kelvin is essential for scientific calculations, whereas Celsius is often more familiar and suitable for everyday interpretations.
monatomic gas
Monatomic gases consist of individual atoms, like helium or neon, and have properties that simplify thermodynamics calculations. For a monatomic ideal gas, the internal energy and temperature are directly related.
  • The equation \( U = \frac{3}{2}nRT \) highlights this relationship.
  • These gases follow simple behavior due to their lack of molecular bonds, unlike diatomic or polyatomic gases.
In solving thermodynamic problems, knowing the gas's atomic property helps determine the number of degrees of freedom, affecting the calculation of internal energy and, subsequently, temperature changes. This is vital in exercises where temperature and energy change are the focus, like in this example with ideal gas conditions.

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Most popular questions from this chapter

Two moles of an ideal gas are compressed in a cylinder at a constant temperature of 65.0\(^\circ\)C until the original pressure has tripled. (a) Sketch a \(pV\)-diagram for this process. (b) Calculate the amount of work done.

Nitrogen gas in an expandable container is cooled from 50.0\(^\circ\)C to 10.0\(^\circ\)C with the pressure held constant at 3.00 \(\times\) 10\(^5\) Pa. The total heat liberated by the gas is 2.50 \(\times\) 10\(^4\) J. Assume that the gas may be treated as ideal. Find (a) the number of moles of gas; (b) the change in internal energy of the gas; (c) the work done by the gas. (d) How much heat would be liberated by the gas for the same temperature change if the volume were constant?

A gas in a cylinder expands from a volume of 0.110 m\(^3\) to 0.320 m\(^3\). Heat flows into the gas just rapidly enough to keep the pressure constant at 1.65 \(\times\) 10\(^5\) Pa during the expansion. The total heat added is 1.15 \(\times\) 10\(^5\) J. (a) Find the work done by the gas. (b) Find the change in internal energy of the gas. (c) Does it matter whether the gas is ideal? Why or why not?

On a warm summer day, a large mass of air (atmospheric pressure 1.01 \(\times\) 10\(^5\) Pa) is heated by the ground to 26.0\(^\circ\)C and then begins to rise through the cooler surrounding air. (This can be treated approximately as an adiabatic process; why?) Calculate the temperature of the air mass when it has risen to a level at which atmospheric pressure is only 0.850 \(\times\) 10\(^5\) Pa. Assume that air is an ideal gas, with \(\Upsilon\) = 1.40. (This rate of cooling for dry, rising air, corresponding to roughly 1 C\(^\circ\) per 100 m of altitude, is called the dry \(adiabatic\) \(lapse\) \(rate\).)

Starting with 2.50 mol of N\(_2\) gas (assumed to be ideal) in a cylinder at 1.00 atm and 20.0\(^\circ\)C, a chemist first heats the gas at constant volume, adding 1.36 \(\times\) 10\(^4\) J of heat, then continues heating and allows the gas to expand at constant pressure to twice its original volume. Calculate (a) the final temperature of the gas; (b) the amount of work done by the gas; (c) the amount of heat added to the gas while it was expanding; (d) the change in internal energy of the gas for the whole process.

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