Physics Problems & Solutions: Temperature

Showing posts with label Temperature. Show all posts

Thermal Physics - Blackbody Radiation

The total energy of a blackbody radiation source is collected for one minute and used to heat water. The temperature of the water increases from 20.0^oCto 20.5^oC. If the absolute temperature of the blackbody is doubled and the experiment repeated, which of the following statements would be most nearly correct?

A. The temperature of the water would increase from 20^oC to 21^oC
B. The temperature of the water would increase from 20^oC to 24^oC
C. The temperature of the water would increase from 20^oC to 28^oC
D. The temperature of the water would increase from 20^oC to 36^oC
E. The water would boil within the one-minute time period.

(GR9277 #14)

Solution:

Blackbody Radiation (Stefan-Boltzmann Law): u = σT⁴ → u ∝ T⁴
u₁∝ T⁴ and u₂∝ (2T)⁴

Heat Transfer: Q = mcΔT → Q ∝ ΔT
Q₁∝ ΔT₁ = 20.5 − 20 = 0.5

u₁/u₂= Q₁/Q₂
T⁴/16T⁴ = 0.5/ΔT₂
ΔT₂= 16 × 0.5 = 8

Answer: C

Thermal Physic - Critical Isotherm

Questions 46-47.

Isotherms and coexistence curves are shown in the PV diagram above for a liquid-gas system. The dashed lines are the boundaries of the labeled regions. Which numbered curve is the critical isotherm?

A. 1
B. 2
C. 3
D. 4
E. 5

(GR9277 #46)

Solution:

Critical isotherm (critical temperature) → dP/dV = 0

the derivative of the curve is zero
the tangent to the curve results in a horizontal line
the point where the vertical and horizontal dashed lines cross

Answer: B

Thermal Physics - Heat Capacity

Questions 71-73

A system in thermal equilibrium at temperature T consists of a large number N₀ of subsystems, each of which can exist only in two states of energy E₁ and E₂, where $E_2-E_1=\epsilon>0$ . In the expressions that follow, k is the Boltzmann constant.

The internal energy of this system at any temperature T is given by $E_1 N_0+\frac{N_0\epsilon}{1+e^{\epsilon/kT}}$ . The heat capacity of the system is given by which of the following expressions?

A. $N_0 k \left (\frac{\epsilon}{kT} \right )^2\frac{e^{\epsilon/kT}}{\left (1+e^{\epsilon/kT} \right )^2}$

B. $N_0 k \left (\frac{\epsilon}{kT} \right )^2\frac{1}{\left (1+e^{\epsilon/kT} \right )^2}$

C. $N_0 k\left( \frac{\epsilon}{kT} \right )^2e^{-\epsilon/kT}$

D. $\frac{N_0 k}{2} \left (\frac{\epsilon}{kT} \right )^2$

E. $\frac{3}{2} N_0 k$

(GR9277 #72)

Solution:

Heat Capacity (at constant volume), $C_V=\left (\frac{\partial U}{\partial T} \right )_V$

$\\U= E_1 N_0+\frac{N_0\epsilon}{1+e^{\epsilon/kT}} \\\\\frac{\partial U}{\partial T}= N_0 k \left (\frac{\epsilon}{kT} \right )^2\frac{e^{\epsilon/kT}}{\left (1+e^{\epsilon/kT} \right )^2}$

Answer: A

Thermal Physics - Blackbody Radiation

A blackbody at temperature T₁ radiates energy at a power level of 10 milliwatts (mW). The same blackbody, when at a temperature 2T₁, radiates energy at a power level of

A. 10 mW
B. 20 mW
C. 40 mW
D. 80 mW
E. 160 mW

(GR8677 #46)

Solution:

Blackbody radiation (Stefan-Boltzmann's law): u = σT⁴

Constant σ = u / T⁴

u₁T₁⁴ = u₂T₂⁴
10 / T₁⁴= u₂/ (2T₁)⁴
u₂ = (10)(2T₁)⁴/ T₁⁴= (10)(16) =160

Answer: E

Thermal Physics - Temperature

For a system in which the number of particles is fixed, the reciprocal of the Kelvin temperature T is given by which of the following derivatives? (Let P = pressure, V = volume, S = entropy, and U = internal energy.)

A. (∂P/∂V)_S
B. (∂P/∂S)_V
C. (∂S/∂P)_U
D. (∂V/∂P)_U
E. (∂S/∂U)_V

(GR8677 #66)

Solution:

The statistical mechanics definition of the temperature:

$T=\left( \frac{\partial S}{\partial U}\right )_{V,N}^{-1}$

The reciprocal of T:

$\frac{1}{T}=\left( \frac{\partial S}{\partial U}\right )_{V,N}$

Answer: E