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Q. 1.23

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An Introduction to Thermal Physics
Found in: Page 17
An Introduction to Thermal Physics

An Introduction to Thermal Physics

Book edition 1st
Author(s) Daniel V. Schroeder
Pages 356 pages
ISBN 9780201380279

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Short Answer

Calculate the total thermal energy in a liter of helium at room temperature and atmospheric pressure. Then repeat the calculation for a liter of air.

Total thermal energy in a liter of helium : 151.98 J

Total thermal energy in a liter of air : 253.31.J

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step1: Given information

Temperature T = 298 K

Degree of freedom of Helium is f=3

Step2: Explanation

Total thermal energy is for a system contains N molecules each with f degree of freedom is given as

Uthermal =Nf12kT ..............................(1)

Where

N = number pf molecules

k = Boltzmann constants

f = degree of freedom

T = Temperature

A liter of helium at room temperature and pressure of atm = 101325 Pa

Using ideal gas law P V=n k T, Find the thermal energy of

Uthermal =3NkT12Uthermal =32PV..............................(2)

Substitute the values given we get

Uthermal =32PV=32×(101325 Pa)×(1×10-3m3)Uthermal =151.98 J

For air , Number of degree of freedom is f=5
Find the thermal energy by substituting values in equation 2.

Uthermal =52PV=52×(101325Pa)×(1×10-3m3)Uthermal =253.31 J

Most popular questions for Physics Textbooks

Consider a narrow pipe filled with fluid, where the concentration of a specific type of molecule varies only along its length (in the x direction). Fick's second law is derived by considering the flux of these particles from both directions into a short segment∆x

∂n∂t=D∂2n∂x2

Given an example to illustrate why you cannot accurately judge the temperature of an object by how hot or cold it feels to the touch?

Consider the combustion of one mole of methane gas:

CH4 (gas) +2O2 (gas) ⟶CO2 (gas) +2H2O( gas )

The system is at standard temperature (298 K) and pressure 105 Pa both before and after the reaction.

(a) First imagine the process of converting a mole of methane into its elemental constituents (graphite and hydrogen gas). Use the data at the back of this book to find ΔHfor this process.

(b) Now imagine forming a mole of CO2 and two moles of water vapor from their elemental constituents. Determine ΔH for this process.

(c) What is ΔHfor the actual reaction in which methane and oxygen form carbon dioxide and water vapor directly? Explain.

(d) How much heat is given off during this reaction, assuming that no "other" forms of work are done?

(e) What is the change in the system's energy during this reaction? How would your answer differ if theH2O ended up as liquid water instead of vapor?

(f) The sun has a mass of2×1030 kg and gives off energy at a rate of 3.9× 1026 watts. If the source of the sun's energy were ordinary combustion of a chemical fuel such as methane, about how long could it last?

Measured heat capacities of solids and liquids are almost always at constant pressure, not constant volume. To see why, estimate the pressure needed to keep V fixed as T increases, as follows.

(a) First imagine slightly increasing the temperature of a material at constant pressure. Write the change in volume,dV1, in terms of d T and the thermal expansion coefficient β introduced in Problem 1.7.

(b) Now imagine slightly compressing the material, holding its temperature fixed. Write the change in volume for this process, d V2, in terms of d P and the isothermal compressibility κT, defined as

κT≡−1V∂V∂PT

(c) Finally, imagine that you compress the material just enough in part (b) to offset the expansion in part (a). Then the ratio of dP to dT is equal to (∂P/∂T)V, since there is no net change in volume. Express this partial derivative in terms of β and κT. Then express it more abstractly in terms of the partial derivatives used to define β and κT. For the second expression you should obtain

∂P∂TV=−(∂V/∂T)P(∂V/∂P)T

This result is actually a purely mathematical relation, true for any three quantities that are related in such a way that any two determine the third.

(d) Compute β,κT, and (∂P/∂T)V for an ideal gas, and check that the three expressions satisfy the identity you found in part (c).

(e) For water at 25∘C,β=2.57×10−4K−1 and κT=4.52×10−10Pa−1. Suppose you increase the temperature of some water from 20∘C to 30∘C. How much pressure must you apply to prevent it from expanding? Repeat the calculation for mercury, for which (at 25∘C ) β=1.81×10−4K−1 andκT=4.04×10−11Pa−1

Given the choice, would you rather measure the heat capacities of these substances at constant v or at constant p ?

Your 200 g cup of tea is boiling-hot. About how much ice should you add to bring it down to a comfortable sipping temperature of 65°C ? (Assume that the ice is initially 65°C. The specific heat capacity of ice is role="math" localid="1650146844935" 0.5 cal/g°C.
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