An ideal monatomic gas is contained in a vessel of constant...

An ideal monatomic gas is contained in a vessel of constant volume 0.200 $\mathrm{m}^{3}$ . The initial temperature and pressure of the gas are $300 . \mathrm{K}$ and 5.00 $\mathrm{atm}$ , respectively. The goal of this problem is to find the temperature and pressure of the gas after 16.0 $\mathrm{kJ}$ of thermal energy is supplied to the gas. (a) Use the ideal gas law and initial conditions to calculate the number of moles of gas in the vessel. (b) Find the specific heat of the gas. (c) What is the work done by the gas during this process? (d) Use the first law of thermodynamics to find the change in internal energy of the gas. (e) Find the change in temperature of the gas. (f) Calculate the final temperature of the gas. (g) Use the ideal gas expression to find the final pressure of the gas.

Key Concepts

Internal Energy of a Monatomic Gas

For an ideal monatomic gas, the internal energy is given by U = (3/2)nRT. This relation shows that the internal energy depends solely on the temperature and the number of moles, assuming the gas behaves ideally. It is a key concept used to relate changes in temperature to changes in internal energy when heat is added.

Work in Thermodynamic Processes

Work in a thermodynamic process is generally calculated by the expression W = ?P dV. In a constant volume process, such as the one described, there is no change in volume (dV = 0), which means that the work done by or on the gas is zero. This concept helps simplify energy calculations using the first law of thermodynamics.

Ideal Gas Law

The Ideal Gas Law (PV = nRT) relates the pressure, volume, temperature, and number of moles of a gas. It is fundamental in thermodynamics for connecting macroscopic state variables and is used to calculate one parameter when the others are known. In this context, it is used to determine the number of moles present in the container given the initial state of the gas.

First Law of Thermodynamics

The First Law of Thermodynamics, expressed as ?U = Q - W, is a statement of energy conservation for thermodynamic systems. It links the change in internal energy (?U) with the heat added to the system (Q) and the work done by the system (W), thereby providing a framework to analyze energy changes during processes such as heating a gas.

Specific Heat Capacity

Specific heat capacity quantifies the amount of heat required to change the temperature of a substance by a unit degree. For an ideal monatomic gas at constant volume, the specific heat capacity is related to the internal energy change, and it plays a critical role in determining the temperature change when a known amount of heat is added.

Transcript

00:01 So here for part a, we know that pv from the ideal gas law, pv equals nrt.

00:06 And so we can say velocity, volume final equals volume initial.

00:12 And this would simply be equalling some variable v volume.

00:15 And this is equalling 0 .2 -0 cubic meters.

00:19 So now we have that the number of moles would be equalling the pressure times the volume divided by the ideal gas constant times the temperature.

00:26 So this would be 5 .0 atmospheres, multiplied by 1 .03 times 10 to the 5th pascal's for every 1 atmosphere, multiplied by the volume of 0 .200 cubic meters.

00:44 And this would all be divided by 8 .31 joules per mole per kelvin, multiplied by 300 kelvin.

00:58 And we find that the number of moles is equaling 40 .6 moles.

01:05 So this would be our final answer for part a.

01:08 For part b, we want to find the total heat capacity, and this would be equalling the number of moles times the molar heat capacity at a constant volume.

01:17 We have a monotomic gas, so this would be n multiplied by 3r over 2.

01:22 So this is equalling 3 over 2, multiplied by 8 .31 joules.

01:31 Joules per mole per kelvin multiplied by 40 .6 moles.

01:39 And we find that here the total heat capacity for the gas would be equaling 506 joules per kelvin.

01:52 Now for part, this would be your answer for part b.

01:55 Now for part c, we have an isochoric process.

02:00 So isochoric meaning constant volume, which simply means that the work done is going to be equaling 0 joules.

02:10 From the first law of thermodynamics for part d, we can say that the change in the internal energy would be equalling the heat plus the work, and this is going to be equaling 16 .0 kilojoules plus 0 joules.

02:28 So we can say then that the change in the internal energy would be equaling positive 6 .5.

02:38 16 .0 kilijoules...

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