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Propane gas (C3H8) behaves like an ideal gas with Υ = 1.127. Determine the molar heat capacity at constant volume and the molar heat capacity at constant pressure.

Short Answer

Expert verified
Cv is approximately 65.45 J/(mol K) and Cp is approximately 73.76 J/(mol K).

Step by step solution

01

Identify Necessary Formulas

We need to determine two values: the molar heat capacity at constant volume, Cv, and the molar heat capacity at constant pressure, Cp. For an ideal gas, the relationship between Cv and Cp is given by the equation Cp=Cv+R, where R is the universal gas constant, approximately 8.314 J/(mol·K).
02

Use Gamma Relation

The heat capacity ratio γ is defined as γ=CpCv. We're given γ=1.127. Using this formula allows us to write Cp=γCv. We will use this to find the individual heat capacities.
03

Set Up Equations

Using the equations from the previous steps, we now have: 1. Cp=γCv 2. Cp=Cv+R.
04

Solve for Cv

Substitute Cp=γCv from Step 3 into the equation Cp=Cv+R: γCv=Cv+R. This simplifies to Cv(γ1)=R.
05

Calculate Cv

Solve for Cv: Cv=Rγ1=8.3141.12718.3140.12765.45 J/(mol·K).
06

Calculate Cp

Now that we have Cv, calculate Cp using Cp=Cv+R: Cp=65.45+8.31473.76 J/(mol·K).

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Key Concepts

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

Molar Heat Capacity
Molar heat capacity refers to the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin). In gases, this quantity is different depending on whether the pressure or volume is held constant during the heating process.
  • At constant volume (Cv): This is the heat capacity when no work is done by or on the gas because the volume of the gas does not change. Here, the energy goes entirely into raising the temperature.
  • At constant pressure (Cp): This is the heat capacity when the gas is allowed to expand, doing work on its surroundings. In this case, both the temperature and the volume of the gas change.
For propane gas, like many other ideal gases, the relationship between the two is given by Cp=Cv+R, where R is the universal gas constant. This relationship underscores the fact that the molar heat capacity at constant pressure is always greater than at constant volume due to the work done during expansion.
Heat Capacity Ratio (Gamma)
The heat capacity ratio, often denoted as γ, is a crucial parameter in thermodynamics, especially concerning gases. It is defined as the ratio of the molar heat capacities:γ=CpCvThis ratio provides insight into how a gas will behave under adiabatic processes (where no heat is exchanged with the surroundings). In the given exercise, γ for propane is 1.127, reflecting its specific properties compared to other gases.
- **Adiabatic Process Insight**: Knowing the value of γ is vital as it affects the speed of sound in a gas and determines the pressure-volume relationship in adiabatic expansions.- **Finding Cv and Cp**: Given γ=1.127, it allows us to use both this relation and the equation Cp=Cv+R to solve for each heat capacity as shown in the step-by-step solution.
Propane Gas Characteristics
Propane (C3H8) is a commonly used hydrocarbon gas that demonstrates behavior similar to an ideal gas under many conditions. Here are some important characteristics of propane:- **Colorless and Odorless**: It's naturally colorless and odorless, but an odorant is usually added for safety reasons.- **Highly Flammable**: Propane is used extensively as a fuel source due to its high energy content.- **Applications**: It is commonly used for heating, cooking, and as a fuel for engines.When considering propane as an ideal gas in exercises, it follows the basic principles of the ideal gas law, allowing us to predict its behavior accurately using equations like the one in the given exercise involving Cp, Cv, γ, and R. Recognizing how these properties drive real-world applications helps understand why equations like the ideal gas law are so useful.

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Most popular questions from this chapter

During an adiabatic expansion the temperature of 0.450 mol of argon (Ar) drops from 66.0C to 10.0C. The argon may be treated as an ideal gas. (a) Draw a pV-diagram for this process. (b) How much work does the gas do? (c) What is the change in internal energy of the gas?

In another test, the valve of a 500-L cylinder full of the gas mixture at 2000 psi (gauge pressure) is opened wide so that the gas rushes out of the cylinder very rapidly. Why might some N2O condense during this process? (a) This is an isochoric process in which the pressure decreases, so the temperature also decreases. (b) Because of the rapid expansion, heat is removed from the system, so the internal energy and temperature of the gas decrease. (c) This is an isobaric process, so as the volume increases, the temperature decreases proportionally. (d) With the rapid expansion, the expanding gas does work with no heat input, so the internal energy and temperature of the gas decrease.

The engine of a Ferrari F355 F1 sports car takes in air at 20.0C and 1.00 atm and compresses it adiabatically to 0.0900 times the original volume. The air may be treated as an ideal gas with Υ = 1.40. (a) Draw a pV-diagram for this process. (b) Find the final temperature and pressure.

In an experiment to simulate conditions inside an automobile engine, 0.185 mol of air at 780 K and 3.00 × 106 Pa is contained in a cylinder of volume 40.0 cm3. Then 645 J of heat is transferred to the cylinder. (a) If the volume of the cylinder is constant while the heat is added, what is the final temperature of the air? Assume that the air is essentially nitrogen gas, and use the data in Table 19.1 even though the pressure is not low. Draw a pV-diagram for this process. (b) If instead the volume of the cylinder is allowed to increase while the pressure remains constant, repeat part (a).

A gas undergoes two processes. In the first, the volume remains constant at 0.200 m3 and the pressure increases from 2.00 × 105 Pa to 5.00 × 105 Pa. The second process is a compression to a volume of 0.120 m3 at a constant pressure of 5.00 × 105 Pa. (a) In a pV-diagram, show both processes. (b) Find the total work done by the gas during both processes.

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