A certain ideal gas has molar heat capacity at constant ume...

A certain ideal gas has molar heat capacity at constant ume $C_{v} .$ 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 2$V_{0}$ and then expands further adiabatically to a final volume of 4$V_{0-}(\text { a) Draw a } p V \text { -diagram for this sequence of processes. }$ (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 $|Q|$ of the total heat flow into or out of the gas for this sequence of processes, and state the direction of heat flow.

Key Concepts

P-V Diagram

The pressure-volume (P-V) diagram is a graphical representation of a thermodynamic process, where pressure is plotted against volume. It is a valuable tool for visualizing the work done by or on a system during different processes. The area under a process curve on a P-V diagram provides a measure of the work involved, making it an essential concept in understanding the performance of thermodynamic cycles and processes.

First Law of Thermodynamics

The first law of thermodynamics is a statement of energy conservation, asserting that the change in the internal energy of a system is equal to the net heat added to the system minus the work done by the system. This principle is fundamental in analyzing various thermodynamic processes, as it provides the framework to account for energy transfers involving work and heat.

Work in Thermodynamic Processes

Work in thermodynamic processes is the energy transferred by a system due to a change in volume. It is determined by integrating the pressure with respect to volume along the process path. The calculation of work varies with the type of process—being simpler under constant pressure (isobaric) and more complex for processes like adiabatic expansion, where pressure and volume are not linearly related.

Heat Capacities

Heat capacities, such as the molar heat capacity at constant volume (Cv) and at constant pressure (Cp), are properties that indicate the amount of heat required to raise the temperature of a substance by one degree under specific conditions. These values are crucial in determining temperature changes and internal energy variations during thermodynamic processes, including isobaric and adiabatic transformations.

Isobaric Process

An isobaric process is one in which the pressure remains constant throughout. In this type of process, when a gas expands or is compressed, the change in volume is directly related to changes in temperature. The work done by the gas is calculated simply as the constant pressure multiplied by the change in volume, and heat transfer is determined by both the change in temperature and the specific heat capacity at constant pressure.

Ideal Gas Law

The ideal gas law (PV = nRT) is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and amount of moles of a gas. It serves as the baseline for understanding the behavior of gases under various conditions and is used to derive relationships between these state variables during different thermodynamic processes.

Adiabatic Process

An adiabatic process is characterized by the absence of heat exchange between the system and its surroundings. In an adiabatic expansion or compression, any change in the system's internal energy is solely due to work done by or on the system. For an ideal gas undergoing this process, the relationship between pressure, volume, and temperature follows the adiabatic condition, often expressed in terms of a specific exponent related to the heat capacities.

Transcript

00:01 Hello everyone here it is given a certain ideal gas has molar heat capacity at constant volume so cv is molar specific heat at constant volume assuming this gas initially occupies a volume v0 so initial volume is v0 at pressure p .0 and temperature t not kelvin the there's expand isobarically so there is a process isobaric expansion so that volume becomes in this process is to be not and then expand further so this is the first process and second process there is an aerobatic expansion so that volume becomes 4b0 we have to draw pv graph for the sequence of process.

02:03 Pv graph, second part, we have to compute the work done by the gas for the sequence of process, work done by the gas.

02:20 See, we have to find the final temperature, that is t3, find the absolute value q, amount of heat transfer in the process.

02:44 Now let us start solving it.

02:52 Now pv graph first pv graph first process is isobaric that is temperature is constant first process is 1 to 2 in which pressure is constant 2 to 3 is is isobaric sorry adiabetic expansion now second part b this is the answer of part a answer of p part we have to calculate work done so work done in the first process initial pressure pressure into change in volume that is p not final volume 2 v0 minus initial volume so it will be p .0 v0 in second process work done cb upon r p2 b2 minus p3 v3 substitute the value so it becomes c b r p2 is p0 2 v0 minus p3 into 4v0 but b3 we don't know so b3 can be calculated here here b3 having the value sorry p3 p2 into v2 b2 upon b1 to the power gamma so it is p1 2 b0 upon 4 b0 to the power gamma substitute this value here.

05:58 So w2 you may write this is your first cv by r to p not b not minus p not 2 to the power minus gamma.

06:30 So total work done in the process w1 plus w2 p.

06:39 Not b0 plus c v .r this is p3 i...

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