River water at a temperature of 290. K is used for cooling in an industrial process. The
water returned to the river is at a temperature of 320. K. What percentage of the river water
can be used for cooling if the water temperature returning to the river is required by law to
increase by no more than 10.0 K? Specific heat of water is 4.184 J / (g K)

To solve this problem, we need to determine the percentage of river water that can be used for cooling if the temperature of the water returning to the river cannot increase by more than 10.0 K.

### Given:
- Initial temperature of river water (\( T_{\text{initial}} \)) = 290 K
- Final temperature of returning water (\( T_{\text{return}} \)) = 320 K
- Maximum allowable temperature increase = 10.0 K
- Specific heat capacity of water (\( c \)) = 4.184 J/g·K

### Approach:

1. **Determine the temperature increase of the returned water**:
   The returned water's temperature increases by:
   \[
   \Delta T = T_{\text{return}} - T_{\text{initial}} = 320 \, \text{K} - 290 \, \text{K} = 30 \, \text{K}
   \]
   
   So, the actual temperature increase is 30 K.

2. **Required maximum temperature increase**:
   By law, the temperature increase is allowed to be no more than 10.0 K.

3. **Calculate the percentage of water that can be used**:
   Let \( x \) be the percentage of river water used for cooling. Since the actual temperature increase is 30 K but the allowable increase is only 10 K, we can set up a proportion:
   \[
   \frac{\Delta T_{\text{allowed}}}{\Delta T_{\text{actual}}} = \frac{x}{100}
   \]
   Substituting values:
   \[
   \frac{10 \, \text{K}}{30 \, \text{K}} = \frac{x}{100}
   \]
   Solving for \( x \):
   \[
   x = \frac{10}{30} \times 100 = 33.33\%
   \]

### Conclusion:
Only **33.33%** of the river water can be used for cooling to comply with the legal requirement that the temperature increase of the returned water does not exceed 10.0 K.

---

Would you like more details or have any questions?

Here are some related questions you might be interested in:
1. How is specific heat capacity related to temperature change?
2. What is the effect of increasing the flow rate of cooling water?
3. How does industrial cooling affect aquatic ecosystems?
4. What are alternative cooling methods that reduce water temperature increases?
5. How do laws regulating water temperature affect industrial processes?

**Tip:** Specific heat capacity measures how much energy is required to change the temperature of a substance by a given amount.

River water at a temperature of 290. K is used for cooling in an industrial process. The water returned to the river is at a temperature of 320. K. What percentage of the river water can be used for cooling if the water temperature returning to the river is required by law to increase by no more than 10.0 K? Specific heat of water is 4.184 J / (g K).

Learn how to calculate the percentage of river water that can be used for cooling in an industrial process. The problem involves understanding heat transfer and proportionality, with key data including the specific heat of water (4.184 J/g·K) and a temperature change constraint of 10 K. This solution is suitable for high school students in Grades 10-12.

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