Question

A particular temperature scale is obtained according to the relation: \[ t = ae^{\alpha} + b \] where \( a \) and \( b \) are constants, and \( t \) is in \( ^\circ {C} \). The thermometric property as measured by the thermometer is \( \alpha \). The values of \( \alpha \) at the ice point and steam point are 6 and 9, respectively. The temperature (in \( ^\circ {C} \)) which gives \( \alpha = 7 \) is ........ {(rounded off to two decimal places).}

Hint:

To solve problems involving unknown constants in exponential equations, use given boundary conditions (e.g., ice point and steam point) to form simultaneous equations and eliminate constants.

Answer 1

We are given the relation: \[ t = ae^\alpha + b \] At ice point (\( t = 0^\circ {C} \), \( \alpha = 6 \)): \[ 0 = ae^6 + b \Rightarrow b = -ae^6 \cdots (1) \] At steam point (\( t = 100^\circ {C} \), \( \alpha = 9 \)): \[ 100 = ae^9 + b \] Substitute equation (1) into the second equation: \[ 100 = ae^9 - ae^6 = a(e^9 - e^6) \Rightarrow a = \frac{100}{e^9 - e^6} \] Now calculate \( a \) and then find \( t \) when \( \alpha = 7 \): \[ a = \frac{100}{e^9 - e^6} \Rightarrow t = ae^7 + b = ae^7 - ae^6 = a(e^7 - e^6) \] \[ t = \frac{100}{e^9 - e^6}(e^7 - e^6) \] Let’s simplify numerically: \[ e^6 \approx 403.43, e^7 \approx 1096.63, e^9 \approx 8103.08 \] \[ t = \frac{100}{8103.08 - 403.43}(1096.63 - 403.43) = \frac{100}{7699.65}(693.20) \approx 9.00^\circ {C} \]