Suppose identical amounts of heat transfer into different masses of copper and water, causing identical changes in temperature. What is the ratio of the mass of copper to water?

To determine the ratio of the mass of copper to the mass of water when identical amounts of heat cause identical changes in temperature, we need to use the concept of specific heat capacity. The formula for heat transfer is:

\[
Q = mc\Delta T
\]

Where:
- \( Q \) is the heat transferred.
- \( m \) is the mass of the substance.
- \( c \) is the specific heat capacity.
- \( \Delta T \) is the change in temperature.

Given that the same amount of heat \( Q \) is transferred into both copper and water, and that both experience the same temperature change \( \Delta T \), we can write:

\[
Q_{\text{copper}} = Q_{\text{water}}
\]

\[
m_{\text{copper}} c_{\text{copper}} \Delta T = m_{\text{water}} c_{\text{water}} \Delta T
\]

Since \( \Delta T \) is the same for both substances, it cancels out, leaving:

\[
m_{\text{copper}} c_{\text{copper}} = m_{\text{water}} c_{\text{water}}
\]

To find the ratio of the masses \( \frac{m_{\text{copper}}}{m_{\text{water}}} \), rearrange the equation:

\[
\frac{m_{\text{copper}}}{m_{\text{water}}} = \frac{c_{\text{water}}}{c_{\text{copper}}}
\]

Using the specific heat capacities:
- \( c_{\text{copper}} \approx 0.385 \, \text{J/g°C} \)
- \( c_{\text{water}} \approx 4.186 \, \text{J/g°C} \)

Substitute these values:

\[
\frac{m_{\text{copper}}}{m_{\text{water}}} = \frac{4.186 \, \text{J/g°C}}{0.385 \, \text{J/g°C}} \approx 10.87
\]

Thus, the ratio of the mass of copper to the mass of water is approximately **10.87:1**.

Would you like any further details or have any questions?

Here are 5 related questions for further exploration:
1. How does the specific heat capacity affect the rate of temperature change in different materials?
2. What are some practical applications of specific heat in engineering?
3. How would the ratio change if the materials were different, like iron and water?
4. What is the physical meaning of a material having a higher specific heat capacity?
5. How can calorimetry be used to experimentally determine the specific heat of a substance?

**Tip:** Remember that materials with lower specific heat capacities will undergo larger temperature changes when the same amount of heat is added compared to materials with higher specific heat capacities!

Discover how to calculate the ratio of the mass of copper to water when identical amounts of heat cause identical temperature changes. Understand the concept of specific heat capacity and its application in determining mass ratios. Suitable for high school students studying thermodynamics.

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