The coefficients of thermal conductivity of copper, mercury and glass are respectively `K_(c ), K_(m) and K_(g)` such that `K_(c ) gt K_(m) gt K_(g)`. If the same quantity of heat is to flow per second per unit area of each and corresponding temperature gradients are `X_(c ), X_(m) and X_(g)`

Options

A. `X_(c ) = X_(m) = K_(g)`
B. `X_(c) gt X_(m) gt X_(g)`
C. `X_(c ) lt X_(m) lt X_(g)`
D. `X_(m) lt X_(c) lt X_(g)`

Answer

To solve the problem, we need to establish the relationship between the temperature gradients \(X_c\), \(X_m\), and \(X_g\) for copper, mercury, and glass, respectively, given that the coefficients of thermal conductivity are \(K_c\), \(K_m\), and \(K_g\) such that \(K_c > K_m > K_g\).

### Step-by-Step Solution:

1. **Understanding Thermal Conductivity**:
The thermal conductivity \(K\) of a material indicates how well it conducts heat. A higher value of \(K\) means the material is a better conductor of heat.

2. **Using Fourier's Law of Heat Conduction**:
According to Fourier's law, the rate of heat transfer \(Q\) through a material is given by:
\[ \frac{Q}{t} = K \cdot A \cdot \frac{\Delta T}{L} \]
where:
- \(Q/t\) is the heat transfer per unit time,
- \(K\) is the thermal conductivity,
- \(A\) is the area through which heat is being transferred,
- \(\Delta T\) is the temperature difference,
- \(L\) is the thickness of the material.

3. **Considering Heat Flow per Unit Area**:
Since we are considering heat flow per second per unit area, we can express it as:
\[ \frac{Q}{A \cdot t} = K \cdot \frac{\Delta T}{L} \]
We denote the temperature gradient as \(X\), which is defined as:
\[ X = \frac{\Delta T}{L} \]
Therefore, we can rewrite the equation as:
\[ \frac{Q}{A \cdot t} = K \cdot X \]

4. **Establishing the Relationship**:
Since the quantity of heat flowing per second per unit area is the same for all three materials, we can set up the following equations:
\[ \frac{Q}{A \cdot t} = K_c \cdot X_c = K_m \cdot X_m = K_g \cdot X_g \]

5. **Inversely Proportional Relationship**:
From the equations above, we can derive that:
\[ X_c = \frac{Q}{A \cdot t K_c}, \quad X_m = \frac{Q}{A \cdot t K_m}, \quad X_g = \frac{Q}{A \cdot t K_g} \]
Since \(K_c > K_m > K_g\), it follows that:
\[ X_c < X_m < X_g \]

6. **Final Conclusion**:
Therefore, the relationship between the temperature gradients is:
\[ X_c < X_m < X_g \]

### Summary:
The temperature gradients are inversely proportional to the thermal conductivities. Since \(K_c > K_m > K_g\), we conclude that \(X_c < X_m < X_g\).