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If \(3.965 \mathrm{m}^{3}\) of methane gas at \(288.75 \mathrm{K}\left(15.6^{\circ} \mathrm{C}\right)\) and 1 atm is equivalent to \(3.785 \mathrm{x}\) \(10^{-3} \mathrm{m}^{3}\) of gasoline as fuel for an automobile engine, what would be volume of the tank required to hold methane at 207 bar and \(288.75 \mathrm{K}\left(15.6^{\circ} \mathrm{C}\right)\) in an amount equivalent to \(37.85 \times 10^{-3} \mathrm{m}^{3}\) of gasoline?

Short Answer

Expert verified
The volume of the tank required is approximately 1.979 m鲁.

Step by step solution

01

Convert Gasoline Volume to Methane Volume

The problem states that 3.965 m鲁 of methane at standard conditions is equivalent to 3.785 脳 10鈦宦 m鲁 of gasoline. First, we find the volume in m鲁 of methane needed for 37.85 脳 10鈦宦 m鲁 of gasoline.To do this, use the relation:\[V_{methane1} = \frac{3.965 \, \text{m}^3}{3.785 \times 10^{-3} \, \text{m}^3} \times 37.85 \times 10^{-3} \, \text{m}^3\]Calculating this gives:\[V_{methane1} = \frac{3.965}{3.785} \times 37.85 \approx 39.695 \, \text{m}^3\]
02

Apply Ideal Gas Law to New Conditions

Now apply the ideal gas law to find the volume at the new conditions. The ideal gas law states:\[PV = nRT\]We will use this formula twice; once for the initial conditions (ambient pressure) and once for the final conditions (207 bar).
03

Calculate Initial Moles of Methane

At 1 atm (101.3 kPa) and 288.75 K, using the ideal gas law:\[P_1V_1 = nRT\]Where \(P_1 = 101.3 \text{ kPa}\), \(V_1 = 39.695 \text{ m}^3\), \(R = 8.314 \text{ J/mol路K}\) and \(T = 288.75 \text{ K}\).Rearranging for \(n\):\[n = \frac{P_1V_1}{RT} = \frac{101.3 \times 39.695}{8.314 \times 288.75} \approx 1701.94 \text{ mol}\]
04

Calculate New Volume at 207 bar

Now use the moles to find the new volume at 207 bar:\[P_2V_2 = nRT\]Solving for \(V_2\):\[V_2 = \frac{nRT}{P_2} = \frac{1701.94 \times 8.314 \times 288.75}{20700} \approx 1.979 \text{ m}^3\]Note: Pressure has been converted to kPa here by multiplying 207 bar by 1000.

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Key Concepts

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

Methane gas
Methane gas is a simple molecule, consisting of one carbon atom bonded to four hydrogen atoms. As one of the principal components of natural gas, methane is commonly used as an energy source. It is both highly combustible and abundant, making it ideal for fuel purposes.
Methane's combustion releases significant energy, which is why it is utilized in internal combustion engines to power vehicles, among other applications. When burned, methane primarily produces carbon dioxide and water, contributing to greenhouse gas emissions, yet it is one of the cleaner-burning fossil fuels available today.
In the context of the exercise, we calculate and utilize methane's volume to find its equivalence as an automobile fuel compared to gasoline. The calculation involves the ideal gas law, which helps determine how much methane is needed under specific conditions to match the energy output of a given volume of gasoline. Understanding how to gauge volume equivalence is crucial when considering methane as an alternative fuel source.
Gasoline equivalence
The concept of gasoline equivalence is essential when comparing different types of fuels. It allows for a common ground of comparison by expressing the energy potential of one type of fuel in terms of another鈥攕pecifically gasoline, which is a standard measure due to its prevalent use in vehicles.
In practical terms, gasoline equivalence is utilized to determine how much of an alternative fuel, like methane, is needed to produce a similar amount of energy as a certain volume of gasoline. In our exercise, we convert the amount of gasoline into its equivalent amount in methane. Through this conversion, we can efficiently calculate the required methane volume needed to achieve similar energy output.
  • This conversion uses a known equivalence ratio between the two fuels.
  • It accounts for factors such as differing energy densities and combustion efficiencies.
  • This is vital when evaluating the viability of methane as an alternative to traditional gasoline in automobiles.
By understanding gasoline equivalence, we can better evaluate alternative fuels not just environmentally but also on how well they can perform in terms of energy production.
Automobile fuel
When considering fuels for automobiles, various factors come into play. This not only includes the energy content of the fuel but how it behaves under different conditions, its emissions profile, and its availability.
Gasoline has traditionally been the go-to automobile fuel due to its high energy density and widespread infrastructure. However, due to environmental concerns and economic factors, there has been an increased interest in alternative fuels like methane, which offers potential environmental benefits and sustainability.
The exercise demonstrated how methane can be used in place of gasoline by illustrating the volume needed under particular conditions. This comprehension helps in understanding:
  • How methane can feasibly power engines traditionally run on gasoline.
  • The adjustments needed in terms of fuel storage and supply technologies.
  • The economic and ecological impacts of transitioning to alternative fuels.
Exploring different automobile fuels like methane not only provides a scientific understanding of chemical equivalences but also paves the way for innovations in vehicle fuel technology.

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Most popular questions from this chapter

For a gas described by the Redlich/Kwong equation and for a temperature greater than \(T_{c},\) develop expressions for the two limiting slopes, $$\lim _{P \rightarrow 0}\left(\frac{\partial Z}{\partial P}\right)_{T} \quad \lim _{P \rightarrow \infty}\left(\frac{\partial Z}{\partial P}\right)_{T}$$ Note that in the limit as \(P \rightarrow 0, V \rightarrow \infty\), and that in the limit as \(P \rightarrow \infty, V \rightarrow \mathrm{b}\).

An ideal gas flows through a horizontal tube at steady state. No heat is added and no shaft work is done. The cross-sectional area of the tube changes with length, and this causes the velocity to change. Derive an equation relating the temperatureto the velocity of the gas. If nitrogen at \(423.15 \mathrm{K}\left(150^{\circ} \mathrm{C}\right)\) flows past one section of the tube at a velocity of \(2.5 \mathrm{m} \mathrm{s}^{-1},\) what is its temperature at another section where its velocity is \(50 \mathrm{m} \mathrm{s}^{-1} ?\) Let \(C_{P}=(7 / 2) R\).

\(\mathbf{A}\) tank of \(0.1-\mathrm{m}^{3}\) volume contains air at \(298.15 \mathrm{K}\left(25^{\circ} \mathrm{C}\right)\) and \(101.33 \mathrm{kPa}\). The tank is connected to a compressed-air line which supplies air at the constant conditions of \(318.15 \mathrm{K}\left(45^{\circ} \mathrm{C}\right)\) and \(1500 \mathrm{kPa} .\) A valve in the line is cracked so that air flows slowly into the tank until the pressure equals the line pressure. If the process occurs slowly enough that the temperaturein the tank remains at \(298.15 \mathrm{K}\left(25^{\circ} \mathrm{C}\right)\), how much heat is lost from the tank? Assume air to be an ideal gas for which \(C_{P}=(7 / 2) R\) and \(C_{V}=(5 / 2) R\).

One mole of air, initially at \(423.15 \mathrm{K}\left(150^{\circ} \mathrm{C}\right)\) and 8 bar, undergoes the following mechanically reversible changes. It expands isothermally to a pressure such that when it is cooled at constant volume to \(323.15 \mathrm{K}\left(50^{\circ} \mathrm{C}\right)\) its final pressure is 3 bar. Assuming air is an ideal gas for which \(C_{P}=(7 / 2) R\) and \(C_{V}=(5 / 2) R,\) calculate \(W, Q, A U,\) and \(A H\).

Develop equations which may be solved to give the final temperature of the gas remaining in a tank after the tank has been bled from an initial pressure \(P_{1}\) to a final pressure \(P_{2}\). Known quantities are initial temperature, tank volume, heat capacity of the gas, total heat capacity of the containing tank, \(P_{1}\), and \(P_{2}\). Assume the tank to be always at the temperature of the gas remaining in the tank, and the tank to be perfectly insulated.

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