Steady-state creep rate data are given in the following table for nickel at \(1000^{\circ} \mathrm{C}(1273 \mathrm{~K})\) : $$ \begin{array}{cc} \hline \dot{\epsilon}_{s}\left(\boldsymbol{s}^{-1}\right) & \boldsymbol{\sigma}[\boldsymbol{M P a}(\boldsymbol{p s i})] \\ \hline 10^{-4} & 15(2175) \\ 10^{-6} & 4.5(650) \\ \hline \end{array} $$ If it is known that the activation energy for creep is \(272,000 \mathrm{~J} / \mathrm{mol}\), compute the steady-state creep rate at a temperature of \(850^{\circ} \mathrm{C}\) (1123 K) and a stress level of \(25 \mathrm{MPa}\) (3625 psi).

Using the given data, we can write the steady-state creep rate equation as: $$ \frac{\dot{\epsilon}_{s1}}{\dot{\epsilon}_{s2}} = \left(\frac{\sigma_1}{\sigma_2}\right)^n $$ where \(\dot{\epsilon}_{s1}\) and \(\dot{\epsilon}_{s2}\) are the steady-state creep rates at stress levels \(\sigma_1\) and \(\sigma_2\) respectively, and n is the stress exponent. Plugging in the provided data from the table: $$ \frac{10^{-4}}{10^{-6}} = \left(\frac{15}{4.5}\right)^n $$ Solving for n would give us the stress exponent.

From Step 1, we now need to solve the equation: $$ \frac{10^{-4}}{10^{-6}} = \left(\frac{15}{4.5}\right)^n $$ Simplifying and going through a logarithm we get: $$ n = \frac{log(100)}{log(\frac{15}{4.5})} $$ Calculate the value of n and save it for the following steps.

The Arrhenius equation is given as: $$ \dot{\epsilon}_{s} = A\sigma^n \exp \left(-\frac{Q}{RT}\right) $$ where \(\dot{\epsilon}_{s}\) is the steady-state creep rate, A is a material constant, σ is the stress level, n is the stress exponent, Q is the activation energy, R is the gas constant (8.314 J/mol·K), and T is the temperature in Kelvin. We know the steady-state creep rates, stress levels, activation energy, and temperature for the given data, along with the stress exponent we found in Step 2. Now we need to find the constant A. Using the data from the table, let's choose the first row and plug in all the values to find A: $$ 10^{-4} = A \times 15^n \times \exp\left(-\frac{272000}{8.314\times1273}\right) $$ Solve for A, and save it for use in the next step.

Now, using the activation energy Q, stress exponent n, and material constant A calculated in previous steps, along with the given temperature T = 1123 K and stress level of \(\sigma = 25 \,\text{MPa}\), we can calculate the steady-state creep rate using the Arrhenius equation: $$ \dot{\epsilon}_{s} = A\sigma^n \exp \left(-\frac{Q}{RT}\right) $$ Plug in all the known values, and solve for \(\dot{\epsilon}_{s}\) (the steady-state creep rate). Once you have the value, you have successfully solved the exercise.