Temperature Variance with Resistivity

[minimario]

Homework Statement

ti[/B]
A resistance thermometer, which measures temperature by measuring the change in resistance of a conductor, is made of platinum and has a resistance of 50.0 Ohms? at 20.0°C. (a) When the device is immersed in a vessel containing melting indium, its resistance increases to 76.8 Ohms?. From this information, find the melting point of indium.

(b) The indium is heated further until it reaches a temperature of 235°C. What is the ratio of the new current in the platinum to the current Imp at the melting point?

Homework Equations

$R = R_0 [1+\alpha(T-T_0)]$
$\alpha_{plat} = 3.92 \cdot 10^{-3}$

The Attempt at a Solution

For part (a), the answer is 157 degrees Celsius. (Says in the book as well)

Now, I'm getting 2 different answers for part (b) when using $T_0 = 20$ and $T_0 = 157$.

When $T_0 = 20$, the final resistance is $50[1+\alpha \cdot 215] = 92.14$

When $T_0 = 157$, the final resistance is $76.8[1+\alpha \cdot 78] = 100.2$

Why the large discrepancy with calculating the same thing? (The 2nd one is correct, per the book)

 

[ehild]

The resistance of the Pt thermometer is almost linear function of the temperature in a wide range.
You have to keep To= 20 C° and Ro=50 Ω, and calculate the resistance from the formula R(T) = Ro(1+α(T-To)).

 

[minimario]

Yes, that is what I did in the first calculation; however, the book performs the second calculation (and obtains diff. answer)

 

[ehild]

When $T_0 = 20$, the final resistance is $50[1+\alpha \cdot 185] = 86.26$

from where is 185 from?

 

[minimario]

I changed it to 215, which is 235 - 20.

 

[minimario]

Anyone can figure out problem?

 

[ehild]

Anyone can figure out problem?

Read my post #2. The coefficient α=3.92⋅10⁻³ °C is valid if To=20 °C, Ro=50Ω. You can not use it for other To-s. The book is wrong.
Think: If the book was right, changing the temperature by a small amount dT and getting the resistance at T+dT, you would have the equation: R(T+dT)= R(T)(1+αdT).
If dt is small, R(T+dT) = R(T) +(dR/dT )dT, and the equation is equivalent to the differential equation dR/dT=αR(T), with the solution R(T)= Roe^αT, an exponential function instead of a linear one.

 

[minimario]

...are you sure? I'm fairly confident it's a decent book :/

 

[ehild]

...are you sure? I'm fairly confident it's a decent book :/

I am sure. The R(T) function can be approximated by a higher-order polynomial in a wider range of temperature if you want more accurate values, but it is not an exponential function. http://www.code10.info/index.php?op...e-thermometers&catid=60:temperature&Itemid=83