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ScanSolve AI Answer
Step 1: Determine the dimensions of the known quantities. Pressure (P): Force per unit area. [P] = ([F])/([A]) = [MLT^-2][L^2] = [ML^-1T^-2] Distance (z): [z] = [L] Boltzmann's constant (k): Energy per unit temperature. Energy is force times distance. [k] = ([Energy])/([Temperature]) = [MLT^-2 · L][] = [ML^2T^-2^-1] Temperature (): [] = [] Step 2: Use the principle of dimensional homogeneity for the exponent. The exponent of e must be dimensionless. [( z)/(k)] = [M^0L^0T^0] Therefore, the dimensions of the numerator must equal the dimensions of the denominator: [][z] = [k][] Substitute the known dimensions: [][L] = [ML^2T^-2^-1][] [][L] = [ML^2T^-2] Solve for []: [] = [ML^2T^-2][L] = [MLT^-2] Step 3: Use the principle of dimensional homogeneity for the main equation. The term ()/() must have the same dimensions as P, because e^-( z)/(k) is a dimensionless quantity. [()/()] = [P] Substitute the dimensions of [] and [P]: [MLT^-2][] = [ML^-1T^-2] Solve for []: [] = [MLT^-2][ML^-1T^-2] [] = [M^1-1L^1-(-1)T^-2-(-2)] [] = [M^0L^2T^0] The dimensions of are [L^2]. [L^2] 3 done, 2 left today. You're making progress.