Coefficient of thermal conductivity is the product of heat, distance and reciprocal of (area x difference in temperature x time). The new value of a unit of coefficient of thermal conductivity, if fundamental units are 21.6 kg, 1 decimetre, 4 K and 1 minute will be ______ `xx 10^(-6)` new units.

Answer

To solve the problem, we need to derive the new unit of the coefficient of thermal conductivity using the given fundamental units. Let's break it down step by step.

### Step 1: Write the formula for the coefficient of thermal conductivity (K)
The coefficient of thermal conductivity (K) is given by the formula:

\[ K = \frac{Q \cdot L}{A \cdot \Delta T \cdot t} \]

where:
- \(Q\) = heat (in joules)
- \(L\) = distance (in meters)
- \(A\) = area (in square meters)
- \(\Delta T\) = change in temperature (in Kelvin)
- \(t\) = time (in seconds)

### Step 2: Identify the units of each quantity
- Heat (Q) has units of energy, which is \( \text{kg} \cdot \text{m}^2/\text{s}^2 \).
- Distance (L) is in meters (m).
- Area (A) is in square meters (m²).
- Change in temperature (\(\Delta T\)) is in Kelvin (K).
- Time (t) is in seconds (s).

### Step 3: Substitute the units into the formula
Substituting the units into the formula for K, we have:

\[ K = \frac{\text{kg} \cdot \text{m}^2/\text{s}^2 \cdot \text{m}}{\text{m}^2 \cdot \text{K} \cdot \text{s}} \]

### Step 4: Simplify the units
Now, simplify the units:

\[ K = \frac{\text{kg} \cdot \text{m}^3/\text{s}^2}{\text{m}^2 \cdot \text{K} \cdot \text{s}} = \frac{\text{kg} \cdot \text{m}}{\text{s}^3 \cdot \text{K}} \]

### Step 5: Substitute the given values
Now we substitute the given fundamental units:
- Mass = 21.6 kg
- Length = 1 decimeter = 0.1 m
- Temperature = 4 K
- Time = 1 minute = 60 seconds

Substituting these values into the formula for K:

\[ K = \frac{21.6 \cdot 0.1}{60^3 \cdot 4} \]

### Step 6: Calculate the denominator
Calculating \(60^3\):

\[ 60^3 = 216000 \]

### Step 7: Substitute and simplify
Now substituting back into the equation:

\[ K = \frac{21.6 \cdot 0.1}{216000 \cdot 4} \]

Calculating the denominator:

\[ 216000 \cdot 4 = 864000 \]

So we have:

\[ K = \frac{2.16}{864000} \]

### Step 8: Final calculation
Now calculating \(K\):

\[ K = \frac{2.16}{864000} = 2.5 \times 10^{-6} \]

### Conclusion
Thus, the new value of the unit of the coefficient of thermal conductivity is:

\[ \boxed{2.5 \times 10^{-6}} \text{ new units} \]