The first law of thermodynamics is an energy conservation law that relates chang

Question

The first law of thermodynamics is an energy conservation law that relates changes in internal energy to energy transfers due to heat and work. If a system undergoes a change from an initial state to a final state, then the change in internal energy U is given by U = Q + W, where Q is energy transferred to the system by heat and W is the work done on the system. The energy transfer Q and work done W between the initial and final states for various processes are summarized in the following tables. Energy Q transferred by heat to a monatomic ideal gas Isobaric process Constant pressure Q = 5 2 nRT = nCpT Adiabatic process No energy enters or leaves the system by heat Q = 0 Isovolumetric process Constant volume Q = nCvT Isothermal process Constant temperature Q = W In the table above for all of these heat transfer energy equations, n is the number of moles of ideal gas, T is the temperature change, Cp and Cv are the molar specific heats at constant pressure and volume respectively, and R is the universal gas constant, R = 8.31 (J/mol · K). Work W done between initial and final states of a monatomic ideal gas Isobaric process Constant pressure W = PV Adiabatic process No energy enters or leaves the system by heat W = U Isovolumetric process Constant volume W = 0 Isothermal process Constant temperature W = nRT ln Vf Vi In the table above for all of these equations for the work done on an ideal gas between the initial and final states, P is the pressure of the ideal gas, n is the number of moles of the gas with initial volume Vi and final volume Vf, and R is the universal gas constant, R = 8.31 (J/mol · K). The Guided Problems guide you through the following Problem-Solving Strategy. • Read the problem carefully at least once. • Identify the thermodynamic process. • Apply the first law of thermodynamics. • Solve for the unknown quantity and substitute the numbers.

Guided Problem

A monatomic ideal gas expands from A to B as in the PV diagram in the figure, where

VA = 2.50 m3,

VB = 3.20 m3,

and

P = 1.40 105 Pa.

If there are 140 moles of gas, determine the following.

(a) the gas temperature at A and B

(b) the work done on the gas

(c) the thermal energy absorbed by the gas

Part 1 of 7

Read the problem carefully at least once.
Be sure to notice the quantities that are known and those quantities that must be found. We are given values for the pressure and volumes of a monatomic ideal gas that expands from state A to state B as shown in the PV diagram. The pressure of the gas is constant such that

P = PA = PB = 1.40 105 Pa.

The volume of the gas in state A is

VA = 2.50 m3

and it expands to a volume of

VB = 3.20 m3

in state B. The problem also states that there are 140 moles of the gas.

The unknown quantities to be determined are (a) the gas temperatures at A and B, (b) the work done on the gas, and (c) the thermal energy absorbed by the gas.

Part 3 of 7

Identify the thermodynamic process. (cont.)
As the gas changes from state A to state B, the pressure is constant so the process is isobaric.

Part (a) can be answered by direct application of the ideal gas law, PV = nRT.

What is the gas temperature TA in state A?

(a) 301°C

(b) 301 K

(c) 385°C

(d) 385 K

(e) none of the above

Part 1 of 7

Read the problem carefully at least once.
Be sure to notice the quantities that are known and those quantities that must be found. The gas is diatomic with an initial volume of 0.408 L, an initial temperature of 318 K and an initial pressure of 3.70 105 Pa. The final volume is 0.468 L.

The unknown quantities to be determined are (a) the final gas pressure and (b) the work done on the gas during the expansion.

What is the initial volume Vi of the gas in m3?

(a) 408 m3

(b) 0.408 m3

(c) 4.08 103 m3

(d) 4.08 104 m3

(e) none of the above

Question 12.1a
A monatomic ideal gas has initial temperature, volume, and pressure given by 345 K, 0.535 m3, and 1.11 105 Pa, respectively. The gas volume is constant and the gas loses 3.10 103 J of energy.

(a) Determine the number of moles n of gas.

__________mol

(b) Determine the final gas temperature Tf.

________ K

Question 12.1b
A monatomic ideal gas is compressed from 0.255 m3 to 0.145 m3 at a constant temperature of 295 K. The initial pressure is 1.10 105 Pa. (Include the sign of the value in your answers.)

(a) Determine the change in internal energy U.

______________ J

(b) Determine the thermal energy Q transferred to the gas.

_____________ J

Explanation / Answer

12.1
a)
Ti = 345 K
Vi = 0.535 m^3
Pi = 1.11 * 10^5 Pa


We know,
PV = n*RT
1.11 * 10^5 * 0.535 = n*8.31 * 345
n = 20.71
No of moles of Gas, n = 20.71 moles

(b)
U = Q + W
For Constant Volume process W = 0
U = Q
As the Gas loses Energy , Q is Negative.

Cv for monatomic ideal gas = 3/2 R
For Constant Volume Process -
Q = nCvT
- 3.10 * 10^3 = 20.71 * 3/2 * 8.31 * T
T = - 12 K
Tf - Ti = -12 K
Tf = Ti - 12 K
Tf = 345 - 12 K
Tf = 333 K
Final Gas Temperature, Tf = 333 K

12.1 b)

(a)
For Isothermal Process =
If the temperature is held constant, the internal energy of the system also is constant, and so U = 0
Change in Internal Energy = 0

(b)
U = Q + W
0 = Q + W
Q = W

Work done = - P*dV
Work done = - 1.10 * 10^5 * (0.145 - 0.255) J
Work done = 12100 J

Q = - 12100 J
Thermal Energy Transferred to the gas, Q = -12100 J

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