Physics quiz question: temperature of plutonium sphere radiating heat

Physics 205A Quiz 7, fall semester 2013
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e Problem 14.65

"Plutonium pellet"
Department of Energy
wki.pe/File:Plutonium_pellet.jpg

An 8.0 kg sample of weapons-grade plutonium radiates approximately 70 watts[*]. The emissivity of plutonium is 0.55[**]. Assume that the environment is 273 K, and that the sample has a surface area of 0.026 m². The temperature of the plutonium sample is:
(A) 550 K.
(B) 740 K.
(C) 820 K.
(D) 4,900 K.

[*] Theodore A. Parish, V. V. Khromov, and Igor Carron, eds., Safety Issues Associated With Plutonium Involvement in the Nuclear Fuel Cycle, Kluwer (1999), p. 81, goo.gl/3XRE5C.
[**] Thermophysical Properties of Materials for Nuclear Engineering: A Tutorial and Collection of Data, International Atomic Energy Agency (2008), p. 36, www-pub.iaea.org/MTCD/publications/PDF/IAEA-THPH_web.pdf.

Correct answer (highlight to unhide): (A)

The power (rate of heat per time) radiated is given by:

Power = –e·σ·A·((T_obj)⁴ – (T_env)⁴),

where a negative value for power (here, "–70 watts") corresponds to a net amount of heat being removed (radiated) from the object, while a positive value corresponds to a net amount of heat being put into (absorbed) by the object (in order to be consistent with the ±Q convention for removing heat from (–) or putting heat into (+) a thermodynamic system). Solving for the temperature T_obj of the plutonium sphere:

((–Power/(e·σ·A)) + (T_env)⁴))^(1/4) = T_obj,

such that:

T_obj = ((–(–70 watts)/((0.55)·(5.670×10^–8 watts/(m²·K⁴))·(0.026 m²)) + (273 K)⁴)^(1/4),

T_obj = (+8.6333419667×10¹⁰ K⁴ + 0.5554571841×10¹⁰ K⁴)^(1/4),

T_obj = (9.1887991508×10¹⁰ K⁴)^(1/4),

T_obj = 550.5727209 K,

or to two significant figures, the temperature of the plutonium sphere is 550 K.

(Response (B) is (Power/(σ·A))^(1/4) + T_env; response (C) is (Power/(e·σ·A))^(1/4) + T_env; response (D) is Power/(e·A).)

Sections 70854, 70855, 73320
Exam code: quiz07b0o7
(A) : 28 students
(B) : 13 students
(C) : 16 students
(D) : 2 students

Success level: 47%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.54