Question 56: Thermodynamics & KTG

Question

A one $\mathrm{kg}$ block of ice at $$-1.5^{\circ} \mathrm{C}$$ falls from a height of $$1.5 \mathrm{~km}$$ and is found melting. The amount of ice melted due to fall, if $$60 \%$$ energy is converted into heat is (Specific heat capacity of ice is $$0.5 \mathrm{~cal} \mathrm{~g}^{-1} \mathrm{~C}^{-1}$$, Latent heat of fusion of ice $$=80 \mathrm{~cal~g}^{-1}$$ )

Accepted Answer

We can approach this problem using the ideal gas equation:
$$PV = nRT$$
Here, $$P$$ is the pressure, $$V$$ is the volume, $$n$$ is the number of moles of the gas, $$R$$ is the gas constant, and $$T$$ is the temperature in Kelvin.
First, let's convert the given quantities into SI units:
Volume $$V = 300 \, \mathrm{cm}^3 = 300 	imes 10^{-6} \, \mathrm{m}^3 = 3.00 	imes 10^{-4} \, \mathrm{m}^3$$
Pressure $$P = 1 \, \mathrm{mm} \, 	ext{of mercury} = 1 \, \mathrm{mmHg}$$
To convert the pressure from mmHg to Pascals (Pa), we use the fact that $$1 \, \mathrm{mmHg} = 133.322 \, \mathrm{Pa}$$:
$$P = 1 \, \mathrm{mmHg} 	imes 133.322 \, \mathrm{Pa}/\mathrm{mmHg} = 133.322 \, \mathrm{Pa}$$
The temperature must be in Kelvin, so we convert from Celsius to Kelvin:
$$T = 27^{\circ} \mathrm{C} = 27 + 273.15 = 300.15 \, \mathrm{K}$$
Substituting these values into the ideal gas equation, we find the number of moles $$n$$:
$$n = \frac{PV}{RT} = \frac{133.322 \, \mathrm{Pa} 	imes 3.00 	imes 10^{-4} \, \mathrm{m}^3}{8.31 \, \mathrm{J} \, \mathrm{mol}^{-1} \, \mathrm{K}^{-1} 	imes 300.15 \, \mathrm{K}}$$
Now, we calculate:
$$n = \frac{133.322 	imes 3.00 	imes 10^{-4}}{\# 8.31 	imes 300.15} = \frac{39.9966 	imes 10^{-4}}{2493.2465} \approx 1.60 	imes 10^{-5} \, \mathrm{mol}$$
To find the number of air molecules, we use Avogadro's number $$N_A$$:
$$N = n 	imes N_A = 1.60 	imes 10^{-5} \, \mathrm{mol} 	imes 6.02 	imes 10^{23} \, \mathrm{molecules/mol}$$
Calculating this, we get:
$$N = 9.63 	imes 10^{18} \, \mathrm{molecules}$$
However, rounding to significant figures (to match the options provided), we get:
$$N \approx 9.65 	imes 10^{18} \, \mathrm{molecules}$$
Thus, the correct answer is:
Option D: $$9.65 	imes 10^{18}$$

Answer

1.69 g

Answer

10 g

Answer

16.9 g

Answer

17.9 g

Thermodynamics & KTG

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