An Introduction to Thermal Physics
1st Edition
ISBN: 9780201380279
Author: Daniel V. Schroeder
Publisher: Addison Wesley
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Chapter 6.2, Problem 18P
To determine
The average value of
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(b) Consider the following heat system on the real line:
U - U = 0, XER, 1>0
%3D
u(x, 0) = | sin x), rER.
i. Use the fundamental solution of the heat equation to write down a solution u to
the system above as an integral.
ii. Show that the solution u that you have found is bounded by 1.
Consider a planetary atmosphere consisting of an ideal gas of
atoms, each of mass m. The temperature T is regarded to be constant, independent of
the height z above ground level. The acceleration of gravity g is constant, and the
pressure at ground level is Po. Find the pressure as a function of z. (Hint: consider a
thin slab of the atmosphere, and write and solve a simple differential equation.)
Using MATLAB editor, make a script m-file which includes a header block and comments:
Utilizing the ideal gas law:
Vmol= RT/P
Calculate the molecular volume where:
R = 0.08206 L-atm/(mol-K)
P = 1.015 atm. and
T = 270 - 315 K in 5 degree increments
Make a display matrix which has the values of T in the first column and Vmol in the second column
Save the script and publish function to create a pdf file from the script in a file named "ECE105_Wk2_L1_Prep_1"
Chapter 6 Solutions
An Introduction to Thermal Physics
Ch. 6.1 - Prob. 2PCh. 6.1 - Prob. 4PCh. 6.1 - Prob. 5PCh. 6.1 - Prob. 6PCh. 6.1 - Prob. 7PCh. 6.1 - Prob. 8PCh. 6.1 - Prob. 9PCh. 6.1 - Prob. 10PCh. 6.1 - Prob. 11PCh. 6.1 - Prob. 12P
Ch. 6.1 - Prob. 13PCh. 6.1 - Prob. 14PCh. 6.2 - Prob. 15PCh. 6.2 - Prob. 16PCh. 6.2 - Prob. 17PCh. 6.2 - Prob. 18PCh. 6.2 - Prob. 19PCh. 6.2 - Prob. 20PCh. 6.2 - For an O2 molecule the constant is approximately...Ch. 6.2 - The analysis of this section applies also to...Ch. 6.3 - Prob. 31PCh. 6.4 - Calculate the most probable speed, average speed,...Ch. 6.4 - Prob. 35PCh. 6.4 - Prob. 36PCh. 6.4 - Prob. 37PCh. 6.4 - Prob. 39PCh. 6.4 - Prob. 40PCh. 6.5 - Prob. 42PCh. 6.5 - Some advanced textbooks define entropy by the...Ch. 6.6 - Prob. 44PCh. 6.7 - Prob. 45PCh. 6.7 - Equations 6.92 and 6.93 for the entropy and...Ch. 6.7 - Prob. 47PCh. 6.7 - For a diatomic gas near room temperature, the...Ch. 6.7 - Prob. 49PCh. 6.7 - Prob. 50PCh. 6.7 - Prob. 51PCh. 6.7 - Prob. 52PCh. 6.7 - Prob. 53P
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- Using the same procedure to determine the fundamental equation of chemical thermodynamics (dG = –SdT + VdP) from the Gibbs free energy of a system (G = H – TS), can you please explain how to find the analogous fundamental equation for (A=U-TS)? Also, can you please handwrite the formula down instead of typing? I get confused with typed formulas sometime.arrow_forwardWhat is the formula for the fluctuation in the (internal) energy of a system Oy? Prove by direct differentiation and using Ə (log Z)| U = kgT² ƏT that the fluctuation in the energy is given by the following formula: of = kgT². ƏT V,Narrow_forwardSoru What is the formula for the fluctuation in the (internal) energy of a system Ou? Prove by direct aifferentiation and using a(log Z)| U = k T². ƏT that the fluctuation in the energy is given by the following formula: 听=kgT2 Ine %3D V.Narrow_forward
- The Joule-Kelvin coefficient ia given by e, V [T (1) Since it involves the absolute temperature T, this relation can be used to deter- mine the absolute temperature T. Consider any readily measurable arbitrary temperature parameter & (e.g., the height of a mercury column). All that is known is that & is some (unknown) function of T; i.e., ở (a) Express (1) in terms of the various directly measurable quantities involving the temperature parameter & instead of the absolute temperature T, i.e., in terms of u' = (08/0p)r, C,' = (đQ/d0), a' = derivative do/dT. (b) Show that, by integrating the resulting expression, one can find T for any given value of d if one knows that & value of o = do at the triple point where T, = 273.16). = 8 (T). V-(0V/80), and the do when T = T. (e.g., if one knowa thearrow_forwardThe Clausius-Clapeyron relation 5.47 is a differential equation that can, in principle, be solved to find the shape of the entire phase-boundary curve. To solve it, however, you have to know how both L and ~V depend on temperature and pressure. Often, over a reasonably small section of the curve, you can take L to be constant. Moreover, if one of the phases is a gas, you can usually neglect the volume of the condensed phase and just take ~V to be the volume of the gas, expressed in terms of temperature and pressure using the ideal gas law. Making all these assumptions, solve the differential equation explicitly to obtain the following formula for the phase boundary curve:This result is called the vapor pressure equation. Caution: Be sure to use this formula only when all the assumptions just listed are valid.arrow_forwardProblem 1: In statistical mechanics, the internal energy of an ideal gas is given by: N. aNkB 2/3 (3NKB U = U(S,V) = е where a is a constant. 1- Show that the variation of the internal energy is given by: 2 dS - \3V 2 dU = dV \3NkB 2- Using the fundamental relation of thermodynamic dU = T.ds – p. dV, show that the equation of state PV = nRT follows from the first expression of U.arrow_forward
- (a) What is the volume occupied by 1.00 mol of an ideal gas at standard conditions—that is, 1.00 atm (= 1.01* 10^5 Pa) and 273 K? (b) Show that the number of molecules per cubic centimeter (the Loschmidt number) at standard conditions is 2.69 *10^9.arrow_forwardTo design a particular kind of refrigerator, we need to know the temperature reduction brought about by adiabatic expansion of the refrigerant gas. For one type of freon, μ = 1.2 K atm-1.What pressure difference is need to provide a temperature drop of 5.0 K? Recall that (∂T/∂p )H= μ and, instead of integrating, just assume that this equation remains true when the infinitesimally small changes (∂x) are replace by finite changes (Δx).arrow_forwardShow that for an ideal gas (@T/ƏV)u = 0, and (@T/ƏP)H= 0.arrow_forward
- When the temperature of liquid mercury increases by one degree Celsius (or one kelvin), its volume increases by one part in 550,000. The fractional increase in volume per unit change in temperature (when the pressure is held fixed) is called the thermal expansion coefficient, β:(where V is volume, T is temperature, and ~ signifies a change, which in this case should really be infinitesimal if {3 is to be well defined). So for mercury, {3 1/550,000 K-1 1.81 X 10-4 K-1. (The exact value varies with temperature, but between 0° C and 200° C the variation is less than 1%(a) Get a mercury thermometer, estimate the size of the bulb at the bottom, and then estimate what the inside diameter of the tube has to be in order for the thermometer to work as required. Assume that the thermal expansion of the glass is negligible.(b) The thermal expansion coefficient of water varies significantly with temperature: It is 7.5 X 10-4 K-1 at 100°C, but decreases as the temperature is lowered until it…arrow_forwardIn a certain physical system, there are two energy states available to a particle: the ground state with energy E₁ = 0 eV, and the excited state with energy E₂ = 1.5 eV. The system is in thermal equilibrium at a temperature T = 300 K. Calculate the Gibbs factor (also known as the Boltzmann factor) for the excited state . Give your answer to two decimal places.arrow_forwardUse statistical thermodynamic arguments to show that for a perfect gas, (∂E/∂V)T = 0.arrow_forward
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