Question

A Carnot engine (E) is working between two temperatures 473K and 273K. In a new system two engines - engine \(E_1\) works between 473K to 373K and engine \(E_2\) works between 373K to 273K. If \(\eta_{12}\), \(\eta_1\) and \(\eta_2\) are the efficiencies of the engines \(E\), \(E_1\) and \(E_2\), respectively, then:

• A. \(\eta_{12}<\eta_1 + \eta_2\)
• B. \(\eta_{12} = \eta_1 + \eta_2\)
• C. \(\eta_{12} = \eta_1 \eta_2\)
• D. \(\eta_{12}>\eta_1 + \eta_2\)

Hint:

Remember, when dealing with multiple Carnot engines operating between different temperature ranges in series, their efficiencies multiply rather than add.

Answer 1

The Correct Option is C

Step 1: Calculate the efficiencies of each engine. The efficiency \(\eta\) of a Carnot engine is given by: \[ \eta = 1 - \frac{T_{\text{cold}}} {T_{\text{hot}}} \] For \(E_1\): \[ \eta_1 = 1 - \frac{373}{473} \] For \(E_2\): \[ \eta_2 = 1 - \frac{273}{373} \]

 Step 2: Calculate the overall efficiency of two engines working in series. The overall efficiency \(\eta_{12}\) when two Carnot engines operate in series is the product of their individual efficiencies, not the sum: \[ \eta_{12} = \eta_1 \times \eta_2 = \left(1 - \frac{373}{473}\right) \times \left(1 - \frac{273}{373}\right) \] This can be simplified and calculated for the exact values. 

Step 3: Verify the relationship. \[ \eta_{12} = \left(1 - \frac{373}{473}\right) \times \left(1 - \frac{273}{373}\right) \] \[ \eta_{12} = \left(\frac{100}{473}\right) \times \left(\frac{100}{373}\right) \] \[ \eta_{12} \approx \left(\frac{100}{473}\right) \times \left(\frac{100}{373}\right) \approx 0.0567 \] Which indicates the multiplicative relationship is correct as per the properties of series Carnot engines.