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Question: What is the mass of one curie of \({{U}^{234}}\)? \(\begin{aligned} & a)3.7\times {{10}^{10}}g...

What is the mass of one curie of U234{{U}^{234}}?
a)3.7×1010g b)3.7×10−10g c)6.25×10−34g d)1.438×10−11g \begin{aligned} & a)3.7\times {{10}^{10}}g \\\ & b)3.7\times {{10}^{-10}}g \\\ & c)6.25\times {{10}^{-34}}g \\\ & d)1.438\times {{10}^{-11}}g \\\ \end{aligned}

Explanation

Solution

One curie is basically defined as 3.7×10103.7\times {{10}^{10}} disintegration or decay per second. So we are asked to calculate the mass of the uranium i.e. U234{{U}^{234}} after 3.7×10103.7\times {{10}^{10}} disintegrations or decay. So to calculate the mass of one curie we need to multiply one atomic mass unit to the number of disintegrations per second.

Complete answer:
It is given in the question that the molar mass of uranium is 234. Hence the mass of an atom (m) of Uranium equals to, m=Molarmass (mass of one mole of uranium)Avogadro’s Number(number of atoms in one mole).....(1)m=\dfrac{\text{Molarmass (mass of one mole of uranium)}}{\text{Avogadro's Number(number of atoms in one mole)}}.....(1).

Avogadro’s number is numerically equal to 6.022×10−236.022\times {{10}^{-23}}. Hence substituting the molar mass of the Uranium and the Avogadro’s number in the above equation 1 we get,

m=2346.022×1023=38.857×10−23amu of Uraniumm=\dfrac{234}{6.022\times {{10}^{23}}}=38.857\times {{10}^{-23}}\text{amu of Uranium}. As per the definition of 1 curie decays take place. Hence the mass of one curie of U234{{U}^{234}}is amu of Uranium times the number of decays, Mathematically written as,

Mass of one curie of Uranium=amu(Uranium) !!×!! No of disintegration taking place in one curie....(2)\text{Mass of one curie of Uranium=amu(Uranium) }\\!\\!\times\\!\\!\text{ No of disintegration taking place in one curie}....\text{(2)}

The atomic mass unit of Uranium is 38.857×10−2338.857\times {{10}^{-23}} and the number of decays in one curie is 3.7×10103.7\times {{10}^{10}}, hence the Mass in one curie of U234{{U}^{234}} from equation 2 we get,

Mass of one curie of Uranium=amu(Uranium) !!×!! No of disintegration taking place in one curie....(2)\text{Mass of one curie of Uranium=amu(Uranium) }\\!\\!\times\\!\\!\text{ No of disintegration taking place in one curie}....\text{(2)}

Mass of one curie of U234=38.857×10−23×3.7×1010=143.7709×10−13, hence Mass of one curie of U234=143.7709 !!×!! 10-13g =1.438 !!×!! 10-11g approx \begin{aligned} & \text{Mass of one curie of }{{U}^{234}}=38.857\times {{10}^{-23}}\times 3.7\times {{10}^{10}}=143.7709\times {{10}^{-13}}\text{, hence} \\\ & \text{Mass of one curie of }{{\text{U}}^{\text{234}}}\text{=143}\text{.7709 }\\!\\!\times\\!\\!\text{ 1}{{\text{0}}^{\text{-13}}}\text{g =1}\text{.438 }\\!\\!\times\\!\\!\text{ 1}{{\text{0}}^{\text{-11}}}\text{g approx} \\\ \end{aligned}

So, the correct answer is “Option D”.

Note:
In the above solution amu is the abbreviation for atomic mass unit. The word curie is denoted by Ci. This is an older practical unit for activity(rate of disintegration or decay)in the honour of MADAME Marie Sklodowska Curie. The standard unit for activity of a radioactive substance is Becquerel(Bq). One Curie is 3.7×10103.7\times {{10}^{10}} times Bq. Curie i.e. Ci was originally defined as “the quantity or the mass of radium emanation in equilibrium with one gram of radium(element). Though curie is not the SI unit it is still used in medicine and other industries due to its practical significance.