To calculate the total number of BTUs required to change 100 lbs of steam at 275°F to 100 lbs of ice at 22°F, we need to account for the following steps:

1. Cool steam from 275°F to 212°F (steam cooling down to the condensation point).
2. Condense the steam at 212°F to water.
3. Cool water from 212°F to 32°F (cooling the water to the freezing point).
4. Freeze the water at 32°F to ice.
5. Cool the ice from 32°F to 22°F (cooling the ice).

Step-by-Step Calculation:

1. Cool the steam from 275°F to 212°F (steam cooling):

The formula for heat transfer is: $Q = m \cdot C_s \cdot \Delta T$ Where:

• $m = 100 \, \text{lbs}$ (mass)
• $C_s = 0.5 \, \text{BTU/lb·°F}$ (specific heat of steam)
• $\Delta T = 275°F - 212°F = 63°F$

$Q_1 = 100 \cdot 0.5 \cdot 63 = 3,150 \, \text{BTUs}$

2. Condense the steam at 212°F to water (latent heat of condensation):

The latent heat of condensation (same as latent heat of boiling) is 970 BTU/lb. $Q_2 = m \cdot \text{latent heat} = 100 \cdot 970 = 97,000 \, \text{BTUs}$

3. Cool the water from 212°F to 32°F (water cooling):

$Q = m \cdot C_w \cdot \Delta T$ Where:

• $C_w = 1.0 \, \text{BTU/lb·°F}$ (specific heat of water)
• $\Delta T = 212°F - 32°F = 180°F$

$Q_3 = 100 \cdot 1.0 \cdot 180 = 18,000 \, \text{BTUs}$

4. Freeze the water at 32°F to ice (latent heat of fusion):

The latent heat of melting (fusion) is 144 BTU/lb. $Q_4 = m \cdot \text{latent heat} = 100 \cdot 144 = 14,400 \, \text{BTUs}$

5. Cool the ice from 32°F to 22°F (ice cooling):

$Q = m \cdot C_i \cdot \Delta T$ Where:

• $C_i = 0.5 \, \text{BTU/lb·°F}$ (specific heat of ice)
• $\Delta T = 32°F - 22°F = 10°F$

$Q_5 = 100 \cdot 0.5 \cdot 10 = 500 \, \text{BTUs}$

Total Heat Transfer:

The total number of BTUs required is the sum of all the steps: $Q_{\text{total}} = Q_1 + Q_2 + Q_3 + Q_4 + Q_5$ $Q_{\text{total}} = 3,150 + 97,000 + 18,000 + 14,400 + 500 = 133,050 \, \text{BTUs}$

So, 133,050 BTUs are required to change 100 lbs of steam at 275°F to 100 lbs of ice at 22°F.

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Would you like more details on any part of the solution?

Here are 5 related questions for further practice:

1. How much heat is required to melt 50 lbs of ice at 32°F?
2. How many BTUs are required to vaporize 200 lbs of water at 212°F?
3. Calculate the energy needed to cool 150 lbs of steam from 300°F to 100°F.
4. How much heat is required to heat 80 lbs of ice at -10°F to water at 50°F?
5. What is the total energy required to convert 100 lbs of ice at -20°F to steam at 300°F?

Tip: Always pay attention to the different stages (phase changes and temperature changes) when calculating heat transfer.