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Magazine Chapter 18: Problem 34
A 0.25 -mol sample of ideal gas initially occupies \(3.5 \mathrm{L}\). If it takes 61 J of work to compress the gas isothermally to 3.0 L, what's the temperature?
Short Answer
Expert verified
The temperature of the gas is around 310.75 K
Step by step solution
01
Find the Universal Gas Constant, R
Use the ideal gas law, which is defined by the formula \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the Universal Gas Constant and T is the temperature. Let's calculate R using the initial values of P, V, n, and T. Assume the initial pressure to be 1 atm. So, \(R = \frac{PV}{nT} \). However, the value of R is a constant, and it is approximately 8.314 J/(mol*K)
02
Finding the Temperature
The work done by the system in an isothermal process can be given by the equation \(W = -nRT \log \left( \frac{V_f}{V_i} \right)\), where \(V_i\) and \(V_f\) are the initial and final volumes, respectively. Here, the work, W, has been given as 61 J (this will be inserted as -61 J as the work is done on the system). We need to find the temperature, T. Plugging in the known values into the equation, we get \(-61 J = -0.25 \text{ moles} * 8.314 J/(mol*K) * T * \log \left( \frac{3.0 \text{ L}}{3.5 \text{ L}} \right)\) Solving this equation for T yields the sought temperature.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Isothermal Process
An isothermal process is a type of thermodynamic process in which the temperature of the system remains constant throughout. This means that while changes in pressure and volume can occur, such as compression or expansion, the temperature (\(T\)) does not change.
In the context of the Ideal Gas Law, which is represented by \(PV = nRT\), the constant temperature in isothermal processes implies that the product of pressure (\(P\)) and volume (\(V\)) remains constant.
In the context of the Ideal Gas Law, which is represented by \(PV = nRT\), the constant temperature in isothermal processes implies that the product of pressure (\(P\)) and volume (\(V\)) remains constant.
- This results in the equation \(P_1V_1 = P_2V_2\), where the subscripts 1 and 2 represent the initial and final states respectively.
- When dealing with ideal gases, the isothermal process is characterized often by means of the characteristic equation for work, \(W = -nRT \log \frac{V_f}{V_i}\), involving natural logarithms.
Universal Gas Constant
The Universal Gas Constant, commonly symbolized as (\(R\)), is a fundamental constant in the Ideal Gas Law. Its value is \(8.314\, ext{J/(mol*K)}\).
This constant relates the energy scale to the temperature scale when dealing with a mole of a substance.
This constant relates the energy scale to the temperature scale when dealing with a mole of a substance.
- The equation \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, and \(T\) is temperature, utilizes (\(R\)) to balance the units and scale these physical properties correctly.
- It combines various measures such as pressure in pascals, volume in liters, moles as the unit for the amount of substance, and temperature in Kelvin.
Work in Thermodynamics
In thermodynamics, work is the energy transferred to or from a system by mechanical means. For gas systems, work is often associated with volume changes.
Specifically, for an ideal gas undergoing an isothermal process, the work done is characterized by the formula \(W = -nRT \log \frac{V_f}{V_i}\).
Specifically, for an ideal gas undergoing an isothermal process, the work done is characterized by the formula \(W = -nRT \log \frac{V_f}{V_i}\).
- This formula shows that work (\(W\)) is related to the number of moles (\(n\)), the gas constant (\(R\)), the temperature (\(T\)), and the ratio of the final and initial volumes (\(V_f\) and \(V_i\)).
- The negative sign indicates that if the gas is being compressed (as indicated by a reduction in volume), the work is done on the system, and therefore the system receives energy.
Moles in Chemistry
A mole is a standard unit of measurement in chemistry that provides a measure of the amount of a substance. A single mole is defined as exactly \(6.022 \times 10^{23}\) particles (Avogadro's number), which can be atoms, molecules, ions, etc.
The mole concept allows chemists to relate macroscopic amounts of material to the inherent atomic and molecular scales.
The mole concept allows chemists to relate macroscopic amounts of material to the inherent atomic and molecular scales.
- For gases, using the Ideal Gas Law, the presence of moles (\(n\)) indicates the quantity of gas being measured.
- The number of moles can be used to predict the volume occupied by a gas under specific conditions, using the relation \(V = nRT/P\) when rearranging the Ideal Gas Law.
