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What mass of this substance must evaporate to freeze 170 g of water initially at 17 °C? (The heat of fusion of water is 334 J/g, and the specific heat of water is 4.18 J/(g⋅K).) Express the mass in grams to two significant digits.

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Final answer:

To calculate the mass of a substance that must evaporate to freeze water, we must sum the energy required to cool the water to 0°C and the heat of fusion to change it to solid, then divide this by the latent heat of vaporization of the evaporating substance.

Step-by-step explanation:

To determine the mass of a substance that must evaporate to freeze 170 g of water initially at 17°C, we first need to calculate the total amount of energy that needs to be removed from the water to turn it into ice at 0°C. This includes the energy needed to cool the water to 0°C and the heat of fusion to change the state from liquid to solid.

To cool the water to 0°C:
Q_cooling = m × c × ΔT
where m = 170 g (mass of the water),
c = 4.18 J/(g·K) (specific heat of water),
ΔT = 17°C (temperature change from 17°C to 0°C).

Q_cooling = 170 g × 4.18 J/(g·K) × 17 K = 12004.2 J

Then, to freeze the water:
Q_freezing = m × L_f
where L_f = 334 J/g (heat of fusion of water).

Q_freezing = 170 g × 334 J/g = 56780 J

The total energy required to freeze the water is the sum of Q_cooling and Q_freezing:
Q_total = Q_cooling + Q_freezing
Q_total = 12004.2 J + 56780 J = 68784.2 J

This energy will come from the latent heat of vaporization of the substance that evaporates. Assuming the latent heat of vaporization (L_v) is given or can be found, we can calculate the mass (m_substance) required to evaporate using the equation:
Q_total = m_substance × L_v

Therefore, the mass that must evaporate is:
m_substance = Q_total / L_v

Since L_v is not provided in this example, the calculation can't be completed without this value. However, once L_v is known, the mass m_substance can be easily calculated and expressed in grams to two significant digits.

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