Question

a nasa spacecraft measures the rate ( r ) at which atmospheric pressure on mars decreases with altitude. the result at a certain altitude is: ( r = 0.0535 , \text{kpa} cdot \text{km}^{-1} )
convert ( r ) to ( \text{kj} cdot \text{m}^{-4} ).

Explanation:

Step1: Recall unit conversion factors

$1\ \text{kPa} = 1000\ \text{Pa}$, $1\ \text{Pa} = \frac{1\ \text{J}}{\text{m}^3}$, $1\ \text{kJ} = 1000\ \text{J}$, $1\ \text{km} = 1000\ \text{m}$

Step2: Convert kPa to kJ/m³

$0.0535\ \text{kPa} = 0.0535 \times 1000\ \text{Pa} = 53.5\ \text{Pa} = 53.5\ \frac{\text{J}}{\text{m}^3} = \frac{53.5}{1000}\ \frac{\text{kJ}}{\text{m}^3} = 0.0535\ \frac{\text{kJ}}{\text{m}^3}$

Step3: Convert per km to per m

$\frac{1}{\text{km}} = \frac{1}{1000\ \text{m}} = 10^{-3}\ \text{m}^{-1}$

Step4: Combine conversions

$R = 0.0535\ \frac{\text{kJ}}{\text{m}^3} \times 10^{-3}\ \text{m}^{-1} = 0.0535 \times 10^{-3}\ \text{kJ·m}^{-4}$

Step5: Simplify the value

$0.0535 \times 10^{-3} = 5.35 \times 10^{-5}$

Answer:

$5.35 \times 10^{-5}$