Day: September 15, 2021

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202109151444 CYU 3.4

When 1.00\,\mathrm{g} of ammonia boils at atmospheric pressure and -33.0\,^\circ\mathrm{C}, its volume changes from 1.47\,\mathrm{cm^3} to 1130\,\mathrm{cm^3}. Its heat of vaporization at this pressure is 1.37\times 10^6\,\mathrm{J/kg}. What is the change in the internal energy of the ammonia when it vaporizes?

Solution.

With L_v representing the latent heat of vaporization, the heat required to vaporize ammonia is

\begin{aligned} Q & =mL_v \\ & =(0.001\,\mathrm{kg})(1.37\times 10^6\,\mathrm{J/kg}) \\ & =1.37\times 10^3\,\mathrm{J} \\ \end{aligned}

Since the pressure on the system is constant at 1.00\,\mathrm{atm}=1.01\times 10^5\,\mathrm{N/m^2}, the work done by ammonia as it is vaporized is

\begin{aligned} W & =p\Delta V \\ & =(1.01\times 10^5\,\mathrm{N/m^2})\big( (1130-1.47)\times (0.01\,\mathrm{m})^3 \big) \\ & = (1.01\times 10^5\,\mathrm{N\, m})(0.00112853) \\ & = 113.98153\,\mathrm{J} \\ \end{aligned}

By the first law of thermodynamics, the internal (/thermal) energy of ammonia during its vaporization changes by

\begin{aligned} \Delta E_{\textrm{int}} & =Q-W \\ & =1.37\times 10^3\,\mathrm{J} - 113.98153\,\mathrm{J}\\ & = 1256.01847\,\mathrm{J} \\ & \approx 1.26\,\mathrm{kJ} \end{aligned}

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202109151432 CYU 1.3

If 25\,\mathrm{kJ} is necessary to raise the temperature of a rock from 25^\circ\mathrm{C} to 30^\circ\mathrm{C}, how much heat is necessary to heat the rock from 45^\circ\mathrm{C} to 50^\circ\mathrm{C}?

Eq. (1.5):

Q=mc\Delta T

Substituting 25\,\mathrm{kJ}=25000\,\mathrm{J} for heat transferred Q and 30\,^\circ\textrm{C} - 25\,^\circ\textrm{C} = 5\,\mathrm{C}^\circ for temperature change \Delta T, we have

mc=\displaystyle{\frac{Q}{\Delta T}=\frac{25000}{5}}=5000\,\mathrm{unit}

To heat the rock from 45\,^\circ\mathrm{C} to 50\,^\circ\mathrm{C}, the heat needed is

\begin{aligned} Q & = mc\Delta T \\ & = (5000)(50-45) \\ & = 25000\,\mathrm{J} \\ & = 25\,\mathrm{kJ} \end{aligned}

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202109151235 CYU 1.7

How does the rate of heat transfer by conduction change when all spatial dimensions are doubled?

Background. (Rate of conductive heat transfer)

The rate of conductive heat transfer through a slab of material is given by P=\displaystyle{\frac{\mathrm{d}Q}{\mathrm{d}t}=\frac{kA(T_{\textrm{h}}-T_{\textrm{c}})}{d}} where P is the power or rate of heat tranfer, A and d are its surface area and thickness, T_{\textrm{h}}-T_{\textrm{c}} is the temperature difference across the slab, and k is the thermal conductivity of the material. More generally, P=-kA\displaystyle{\frac{\mathrm{d}T}{\mathrm{d}x}} where x is the coordinate in the direction of heat flow.

\begin{aligned} A_{\textrm{initial}} & = l^2 \\ A_{\textrm{final}} & = l'^2 \\ (\,\dots\enspace \textrm{as } l' & =2l\enspace \dots\, ) \\ A_{\textrm{final}} & = (2l)^2 \\ & = 4l^2 \\ & = 4A_{\textrm{initial}} \end{aligned}

and

d_{\textrm{final}}=2d_{\textrm{initial}},

so,

\begin{aligned} P_{\textrm{final}} & = \frac{kA_{\textrm{final}}(T_{\textrm{h}}-T_{\textrm{c}})}{d_{\textrm{final}}} \\ & = \frac{k(4A_{\textrm{initial}})(T_{\textrm{h}}-T_{\textrm{c}})}{(2d_{\textrm{initial}})} \\ & = 2\cdot \frac{kA_{\textrm{initial}}(T_{\textrm{h}}-T_{\textrm{c}})}{d_{\textrm{initial}}} \\ & = 2P_{\textrm{initial}} \end{aligned}

\therefore The rate of heat transfer by conduction increases by a factor of two.

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202109151148 CYU 1.9

How much greater is the rate of heat radiation when a body is at the temperature 40\,^\circ\mathrm{C} than when it is at the temperature 20\,^\circ\mathrm{C}?

Background. (Stefan-Boltzmann law of radiation)

P=\sigma AeT^4

where \sigma = 5.67\times 10^{-8}\,\mathrm{J/s}\cdot\mathrm{m^2}\cdot\mathrm{K^4} is the Stefan-Boltzmann constant; A is the surface area of the object; and T is its temperature in kelvins.

By temperature conversion formula,

T_{\mathrm{(K)}}=T_{\mathrm{(^\circ C)}}+273.15.

\begin{aligned} & T_{\mathrm{(^\circ C)}} = 20 \\ \Rightarrow \qquad & T_{\mathrm{(K)}} = 20 + 273.15 = 293.15 \\ & T_{\mathrm{(^\circ C)}} = 40 \\ \Rightarrow \qquad & T_{\mathrm{(K)}} = 40 + 273.15 = 313.15 \\ \end{aligned}

The rate of heat transfer by emitted radiation is described by the Stefan-Boltzmann law of radiation:

P=\sigma AeT^4.

When the temperature increases from 20\,^\circ\mathrm{C} to 40\,^\circ\mathrm{C}, ceteris paribus, the percentage change of rate of heat radiation is

\begin{aligned} & \qquad \frac{P_{\textrm{new}}-P_{\textrm{old}}}{P_{\textrm{old}}} \times 100 \% \\ & = \frac{\sigma AeT_{\textrm{new}}^4-\sigma AeT_{\textrm{old}}^4}{\sigma AeT_{\textrm{old}}^4} \times 100 \% \\ & = \frac{T_{\textrm{new}}^4-T_{\textrm{old}}^4}{T_{\textrm{old}}^4} \times 100 \% \\ & = \frac{313.15^4-293.15^4}{293.15^4}\times 100 \% \\ & \approx 30.2 \% \qquad \textrm{(3\, s.f.)} \\ \end{aligned}

\therefore The rate of heat transfer increases by about 30\% of the original rate.