Question

Newton's law of cooling leads to the expression: 

Options

• (θ − θ₀) = Kt + c

• log (θ − θ₀) = −Kt + c

• log θ₀ = Kt + c

• θ = Kθ₀ + c

Answer 1

(θ − θ₀) = Kt + c

Explanation:

According to Newton's law of cooling,

`("d"θ)/"dt" K (θ - θ_0)`

where K is the constant of proportionality. Integrating equation (i), we get

`int_theta^(theta_0) (d theta)/(theta - theta_0) = int_0^"t" "Kdt"`

∴ `-int_theta^(theta_0) (d theta)/(theta - theta_0)= int_0^"t" "Kdt"`

∴ `-int_theta^(theta_0) (d theta)/(theta - theta_0) = -int_0^"t" "Kdt"`

∴ log_e = (θ - θ₀) = −Kt + c

∴ 2.303 log₁₀ (θ - θ₀) = −Kt + c

As 2.303 is constant, above equation can be equated to,

log (θ − θ₀) = Kt + c