Question

Consider the (nondimensionalized) ideal gas law for the temperature as a function of pressure (P), volume (V) and mass (M). T(P,V,M)= pv/m

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Find the linearization L(P,V,M) of T(P,V,M) about the point (P,V,M)=(2,3,1). (b) (5pts.) Use differentials to determine approximately how much the temperature ΔT would change if the values of P,V,M in part (a) changed by ΔP= 1/2 ,ΔM=− 1/2 and ΔV=0.

Answer 1

To find the linearization of the temperature function T(P,V,M) about the point (2,3,1), use partial derivatives. To determine the approximate change in temperature, use differentials.

Step-by-step explanation:

To find the linearization of the temperature function T(P,V,M) about the point (2,3,1), we can use partial derivatives. The linearization L(P,V,M) of T(P,V,M) is given by:

L(P,V,M) = T(P₀,V₀,M₀) + (∂T/ ∂P)(P-P₀) + (∂T/ ∂V)(V-V₀) + (∂T/ ∂M)(M-M₀)

where P₀ = 2, V₀ = 3, M₀ = 1, and (∂T/ ∂P), (∂T/ ∂V), and (∂T/ ∂M) are the partial derivatives of T(P,V,M) with respect to P, V, and M, respectively. Plug in the values and calculate L(P,V,M).

(b) To determine the approximate change in temperature ΔT, we can use differentials. ΔT ≈ (∂T/ ∂P)(ΔP) + (∂T/ ∂V)(ΔV) + (∂T/ ∂M)(ΔM). Plug in the values ΔP = 1/2, ΔV = 0, and ΔM = -1/2 along with the partial derivatives calculated in part (a) to find ΔT.