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Propane gas (\(C_3H_8\)) behaves like an ideal gas with \(_\Upsilon\) = 1.127. Determine the molar heat capacity at constant volume and the molar heat capacity at constant pressure.

Short Answer

Expert verified
\(C_v\) is approximately 65.45 J/(mol K) and \(C_p\) 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, \(C_v\), and the molar heat capacity at constant pressure, \(C_p\). For an ideal gas, the relationship between \(C_v\) and \(C_p\) is given by the equation \(C_p = C_v + R\), where \(R\) is the universal gas constant, approximately 8.314 J/(mol·K).
02

Use Gamma Relation

The heat capacity ratio \(\gamma\) is defined as \(\gamma = \frac{C_p}{C_v}\). We're given \(\gamma = 1.127\). Using this formula allows us to write \(C_p = \gamma C_v\). 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. \(C_p = \gamma C_v\) 2. \(C_p = C_v + R\).
04

Solve for \(C_v\)

Substitute \(C_p = \gamma C_v\) from Step 3 into the equation \(C_p = C_v + R\): \(\gamma C_v = C_v + R\). This simplifies to \(C_v(\gamma - 1) = R\).
05

Calculate \(C_v\)

Solve for \(C_v\): \[ C_v = \frac{R}{\gamma - 1} = \frac{8.314}{1.127 - 1} \approx \frac{8.314}{0.127} \approx 65.45 \text{ J/(mol·K)}. \]
06

Calculate \(C_p\)

Now that we have \(C_v\), calculate \(C_p\) using \(C_p = C_v + R\): \[ C_p = 65.45 + 8.314 \approx 73.76 \text{ J/(mol·K)}. \]

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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 (\(C_v\)): 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 (\(C_p\)): 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 \(C_p = C_v + R\), 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 \(\gamma\), is a crucial parameter in thermodynamics, especially concerning gases. It is defined as the ratio of the molar heat capacities:\[ \gamma = \frac{C_p}{C_v} \]This ratio provides insight into how a gas will behave under adiabatic processes (where no heat is exchanged with the surroundings). In the given exercise, \(\gamma\) for propane is 1.127, reflecting its specific properties compared to other gases.
- **Adiabatic Process Insight**: Knowing the value of \(\gamma\) is vital as it affects the speed of sound in a gas and determines the pressure-volume relationship in adiabatic expansions.- **Finding \(C_v\) and \(C_p\)**: Given \(\gamma = 1.127\), it allows us to use both this relation and the equation \(C_p = C_v + R\) to solve for each heat capacity as shown in the step-by-step solution.
Propane Gas Characteristics
Propane (\(C_3H_8\)) 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 \(C_p\), \(C_v\), \(\gamma\), 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

A cylinder with a frictionless, movable piston like that shown in Fig. 19.5 contains a quantity of helium gas. Initially the gas is at 1.00 \(\times\) 10\(^5\) Pa and 300 K and occupies a volume of 1.50 L. The gas then undergoes two processes. In the first, the gas is heated and the piston is allowed to move to keep the temperature at 300 K. This continues until the pressure reaches 2.50 \(\times\) 10\(^4\) Pa. In the second process, the gas is compressed at constant pressure until it returns to its original volume of 1.50 L. Assume that the gas may be treated as ideal. (a) In a \(pV\)-diagram, show both processes. (b) Find the volume of the gas at the end of the first process, and the pressure and temperature at the end of the second process. (c) Find the total work done by the gas during both processes. (d) What would you have to do to the gas to return it to its original pressure and temperature?

A certain ideal gas has molar heat capacity at constant volume \(C_V\) . A sample of this gas initially occupies a volume \(V_0\) at pressure \(p_0\) and absolute temperature \(T_0\) . The gas expands isobarically to a volume \(2V_0\) and then expands further adiabatically to a final volume \(4V_0\) . (a) Draw a \(pV\)-diagram for this sequence of processes. (b) Compute the total work done by the gas for this sequence of processes. (c) Find the final temperature of the gas. (d) Find the absolute value of the total heat flow \(Q\) into or out of the gas for this sequence of processes, and state the direction of heat flow.

During an adiabatic expansion the temperature of 0.450 mol of argon (Ar) drops from 66.0\(^\circ\)C to 10.0\(^\circ\)C. 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 a test of the effects of low temperatures on the gas mixture, a cylinder filled at 20.0\(^\circ\)C to 2000 psi (gauge pressure) is cooled slowly and the pressure is monitored. What is the expected pressure at -5.00\(^\circ\)C if the gas remains a homogeneous mixture? (a) 500 psi; (b) 1500 psi; (c) 1830 psi; (d) 1920 psi.

Three moles of an ideal monatomic gas expands at a constant pressure of 2.50 atm; the volume of the gas changes from 3.20 \(\times\) 10\(^{-2}\) m\(^3\) to 4.50 \(\times\) 10\(^{-2}\) m\(^3\). Calculate (a) the initial and final temperatures of the gas; (b) the amount of work the gas does in expanding; (c) the amount of heat added to the gas; (d) the change in internal energy of the gas.

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