Question

Part 2 (This problem involves concepts from Section 13.2 and 13.3 of the text, which are part of this week’s assigned reading.) Suppose you have two objects in contact, so they can exchange thermal energy, but isolated from the rest of the world. Let the initial temperature of one of them (the hot one) be Th, and its heat capacity Ch, and let the initial temperature of the other one (the cold one) be Tc, and its heat capacity Cc. After a while, they both reach thermal equilibrium at a common temperature Tf . (a) Write an expression, involving the quantities given, for the heat given off by the initially hotter object in the process of reaching thermal equilibrium. (b) Write the corresponding expression for the heat taken in by the initially colder object. (c) According to the law of conservation of energy, what relationship has to hold between these two quantities? (d) Use the result of part (c) to derive an expression for the system’s final temperature, Tf .

Answer 1

m_h = mass of hot one

m_c = mass of cold one

T_h = initial temperature of hot one

T_c = initial temperature of cold one

T_f = final equilibrium temperature

(a)

Heat given off by the hot object is given as

Q_h = m_h C_h (T_h - T_f )

(b)

Heat taken inf by the cold object is given as

Q_c = m_c C_c (T_f - T_c )

c)

as per conservation of energy

Heat lost by the hot object = heat gained by the cold object

Q_h = Q_c

d)

Q_h = Q_c

m_h C_h (T_h - T_f ) = m_c C_c (T_f - T_c )

m_h C_h T_h - m_h C_h T_f = m_c C_c T_f - m_c C_c T_c

(m_h C_h T_h + m_c C_c T_c ) = (m_c C_c T_f + m_h C_h T_f )

T_f = (m_h C_h T_h + m_c C_c T_c )/(m_c C_c + m_h C_h )